Semiconductor Electronic: Material, Devices And Simple Circuits Class 12 Notes Class 12 Notes Chapter 14

1. Metals They possess very low resistivity or high conductivity.
ρ ~ 10^-2.10^-8 Ωm, σ ~10². 108 Sm^-1
2. Semiconductors They have resistivity or conductivity intermediate to metals and insulators.
ρ ~ 10^-5. 10⁶ Ωm, σ ~ 10⁺⁵ .10^-6 Sm^-1
Types of Semiconductors Types of semiconductors are given below:
(i) Elements Semiconductors These semiconductors are available in natural form, e.g. silicon and germanium.
(ii) Compound Semiconductors These semiconductors are made by compounding the metals, e.g. CdS, GaAs, CdSe, InP, anthracene, polyaniline, etc.
3. Insulators They have high resistivity or low conductivity.
ρ ~ 10¹¹ . 10¹⁹ Ωm, σ ~ 10^-11. 10^-19 Sm^-1
4. Energy Band In a crystal due to interatomic interaction, valence electrons of one atom are shared by more than one atom in the crystal. Now, splitting of energy level takes place. The collection of these closely spaced energy levels are called an energy band.
5. Valence Band Valence band are the energy band which includes the energy levels of the valence electrons.
6. Conduction Band Conduction band is the energy band above the valence band.
7. Energy Band Gap The minimum energy required for shifting electrons from valence band to conduction band is called energy band gap (E_g ).
8. Differences between conductor, insulator and semiconductor on the basis of energy bands are given below:

9. Fermi Energy It is the maximum possible energy possessed by free electrons of a material at absolute zero temperature (i.e. 0K)
10. On the basis of purity , semiconductors are of two types:
(i) Intrinsic Semiconductors It is a pure semiconductor without any significant dopant species present
n_e  = n_h =n_i 
where , n_e and n_h are number densities of electrons and holes respectively and n_i  is called intrinsic carrier concentration.
An intrinsic semiconductor is also called an undoped semiconductor or i-type semiconductor
(ii) Extrinsic Semiconductors Pure semiconductor when doped with the impurity, it is known as extrinsic semiconductor.
Extrinsic semiconductors are basically of two types: (a) n-type semiconductors
(b) p-type semiconductors
NOTE: Both the type of semiconductors are electrically neutral.
11. In n-type semiconductor, majority charge carriers are electrons and minority charge carriers are holes, i.e. n_e> n_h .
Here, we dope Si or Ge with a pentavalent element, then four of its electrons bond with the four silicon neighbours, while fifth remains very weakly bound to its parent atom.
Formation of n-type semiconductor is shown below:

12. In p-type semiconductor, majority charge carriers are holes and minority charge carriers are eletron i.e. n_h  > n_e .
In a p-type semiconductor, doping is done with trivalent impurity atoms, i.e. those atoms which have three valence electrons in their valence shell.
Formation of p-type semiconductor is shown below:

13. At equilibrium condition, n_e n_h = n_i²
14. Minimum energy required to create a hole-electron pair, hv > E_g  where, E_g  is energy band gap.
15. Electric current, I = eA(n_ev_e + n_hv_h) where, A is area of cross-section.
where, v_e and v_h are speed of electron and hole respectively.

18. p-n Junction A p-n junction is an arrangement made by a close contact of n-type semiconductor and p-type semiconductor.
19. Formation of Depletion Region in p-n Junction During formation of p-n junction, due to the concentration gradient across p and n sides, holes diffuse from p-side to n-side (p —> n) and electrons diffuse from n-side to p-side (n —> p).
This space charge region on either side of the junction together is known as depletion region.
Depletion region is free from mobile charge carriers. Width of depletion region is of the order of 10^-6 m. The potential difference developed across the depletion region is called the potential barrier.

20. Semiconductor Diode/p-n Junction Diode A semiconductor diode is basically a p-n junction with metallic contacts provided at the ends for the application of an external voltage.

The direction of arrow indicates the conventional direction of current (when the diode is under forward bias).
21. The graphical relations between voltage applied across p-n junction and current flowing through the junction are called I-V characteristics of junction diode.
22. (i) Junction diode is said to be forward bias when the positive terminal of the external
battery is connected less to the p-side and negative terminal to the n-side of the diode. The circuit diagram and I-V characteristics of a forward biased diode is shown below:

The circuit diagram and I-V characteristics of a reverse biased diode is shown below.

23. The DC resistance of a junction diode,
r_DC = V/I
24. The dynamic resistance of junction diode,
r_AC = ∆V/∆I
25. Diode as Rectifier The process of converting alternating voltage/current into direct voltage/current is called rectification. Diode is used as a rectifier for converting alternating current/voltage into direct current/voltage.
There are two ways of using a diode as a rectifier i.e.

(i) Diode as a Half-Wave Rectifier Diode conducts corresponding to positive half cycle and does not conduct during negative half cycle. Hence, AC is converted by diode into unidirectional pulsating DC. This action is known as half-wave rectification.
Circuit diagram of p-n junction diode as half-wave rectifier is shown below:
The input and output wave forms have been given below:

(ii) Diode as a Full-Wave Rectifier In the full-wave rectifier, two p-n junction diodes, D₁ and D₂ are used. The circuit diagram or full-wave rectifier is shown below:

The input and output wave forms have been given below:

Its working based on the principle that junction diode offer very low resistance in forward bias and very high resistance in reverse bias.
26. The average value or DC value obtained from a half-wave rectifier,

27. The average value or DC value obtained from a full-wave rectifier,

28. The pulse frequency of a half-wave rectifier is equal to frequency of AC.
29. The pulse frequency of a full-wave rectifier is double to that of AC.
30. Optoelectronic Devices Semiconductor diodes in which carriers are generated by photons, i.e. photo-excitation, such devices are known as optoelectronic devices.
These are as follows:
(i) Light Emitting Diode (LED) It is a heavily doped p-n junction diode which converts electrical energy into light energy.

LEDs has the following advantages over conventional incandescent low power lamps
(a) Fast action and no warm up time required
(b) It is nearly monochromatic
(c) Low operational voltage and less power consumed
(d) Fast ON-OFF switching capability.
(ii) Photodiode A photodiode is a special type of junction diode used for detecting optical signals. It is a reverse biased p-n junction made from a photosensitive material. Its symbol is

Its V-I characteristics of photodiode are shown below:

We observe from the figure that current in photodiode changes with the change in light intensity (I) when reverse bias is applied.
(iii) Solar Cell Solar cell is a p-n junction diode which converts solar energy into electrical energy. Its symbol is

V-I characteristics of solar cell are shown below:

The materials used for solar cell are Si and GaAs.
31. Zener Diode Zener diode is a reverse biased heavily doped p-n junction diode. It is operated in breakdown region.

32. Zener Diode as a Voltage Regulator When the applied reverse voltage (V) reaches the breakdown voltage (V_z) of the Zener diode there is a large change in the current. So, after the breakdown voltage V_z, a large change in the current can be produced by almost insignificant change in the reverse bias voltage i.e. Zener voltage remains constant even though the current through the Zener diode varies over a wide range. The circuital arrangement is shown as follows.

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Communication Systems Class 12 Notes Chapter 15

Topic 1 Communication

1. Communication Communication is the act of transmission and reception of information.
2. Communication System A system comprises of transmitter, communication channel and receiver.
A block diagram of a generalised communication system is shown as below:

3. Transmitter It consists of transducer/signal generators, modulators and transmitting antenna.
4. Receiver Its main function is to decode the original signals. The main function involves picking up the signals, demodulating and displace the original message signal.
5. Communication Channel The physical path between the transmitter and receiver is known as communication channel. They are of two types namely
(i) Guided (point-to-point) (ii) Unguided
6. Bandwidth of Communication Channel The range of frequencies used to pass through channel is known as bandwidth.

7. The following table shows the various things used in communication system.

8. There are two basic modes of communication given as below:
(i) Point-to-point In this type of communication mode, communication takes place over a link between a signal transmitter and a receiver, e.g. telephony.
(ii) Broadcast In the broadcast mode, there are a large number of receivers corresponding to a signal transmitter, e.g. radio and TV.
9. Basic Terminology used in Electronic Communication Systems
(i) Signal Information converted into electrical form and suitable for transmission is called a signal.
(ii) Transducer Any device/arrangement that converts one form of energy into another is called a transducer, e.g. microphone.
(iii) Noise It refers to the unwanted signals that tends to disturb the transmission and processing of message signals in communication system.
(iv) Attenuation It refers to the loss of strength of a signal during its propagation through the communication channel.
(v) Amplification It is the process by which amplitude of a signal is increased using an electronic circuit called the amplifier.
(vi) Range It is the largest distance between a source and a destination up to which the signal is received with sufficient strength.
(vii) Baseband Band of frequencies representing the original signal is called baseband.
(viii) Repeater Repeaters are erected at suitable distances between the transmitter and receiver. Repeaters are used to extend the range of a communication system.
10. Message Signals A time varying electrical signal generated by a transducer out of original signal is termed as message signal.
The electrical signals are of two types such as below:
(i) Analog signal A continuous signal value which at any instant lies within the range of a maximum and a minimum value.
Graphical representation of analog signal can be represented as given below:

(ii) Digital Signal (Pulse Signal) Digital signals are those which can take only discrete stepwise values e.g. output of a computer, fax, etc.

11. Coding schemes used for digital communication are given as below:
(i) Binary Coded Decimal (BCD) In this, a digit is represented by two binary numbers 0 or 1.
(ii) American Standard Code for Information Interchange (ASCII) It is a universally popular digital code to represent numbers, letters and certain characters.
12. Bandwidth of Signals Bandwidth of signal is defined as the difference between the upper and lower frequencies of signal. In a communication system, the message signal can be voice, music, picture or computer data. This has been shown in the table given as below:

13. Bandwidth of Transmission Medium The commonly used transmission media are wire, free space, fibre optic cable (750 MHz ) and optical fibre (100 GHz.)
This range is sub-divided further and allocated for various services as indicated in the table given as below:

14. Antenna Antenna is a device which acts as an emitter of electromagnetic waves and it also acts as a first receiver of energy. It is generally a metallic object often a wire or collection of wires.
(i) Hertz Antenna It is a straight conductor of length equal to half the wavelength of radio signals to be transmitted or received.
i.e, l = λ/2
(ii) Marconi Antenna It is a straight conductor of length equal to a quarter of the wavelength of radio signals to be transmitted of received, i.e. l = λ/4
(iii) Dipole Antenna It is used in transmission of radio waves. It is omni directional.
(iv) Dish-Type Antenna It is a directional antenna. Such antenna has a parabolic reflector with an active element.
15. Propagation of Electromagnetic Waves In communication using radio waves, an antenna at the transmitter radiates the EM waves, which travel through the space and reach the receiver at the other end.
16. Depending upon frequency and ways of propagation, electromagnetic waves categorised as follows
(i) Ground Wave Propagation (f< 2MHz) In ground wave propagation, the radio waves (AM) travel along the surface of the earth. These waves are guided along the earth surface and they follow the curvature of the earth.

(ii) Sky Wave Propagation (2 MHz < f < 30 MHz) Long distance communication can be achieved by ionospheric reflection of radio waves back towards earth. This mode of propagation is called sky wave propagation and is used by short wave broadcast services. The ionosphere is so called because of the presence of a large number of ions. It extends from height of 65 km to about 400 km above the earth’s surface.
The details are in the table as below:

The density of atmosphere decreases with height.
The ionospheric layer acts as a reflector for a certain range of frequencies. These phenomena are shown as below:

(a) Maximum Usable Frequency (MUF) It is a limiting frequency, but for some specific angle of incidence other than the normal and is given by
MUF = f_c secθ
where, θ is the angle between normal and the direction of incidence of waves.
(b) Skip Distance It is the shortest distance from a transmitter measured along the surface of earth at which a sky wave of fixed frequency c more than f_c will be returned to earth.

(c) Critical Frequency For a given layer, it is the highest frequency that will return down to earth by that layer.

(iii) Space Wave Propagation (LoS) (f > 30 MHz) A space wave travels in a straight line from transmitting anteiina to the receiving antenna.
Space waves are used for Line-of-Sight (LoS) communication as well as satellite communication.
Because of LoS nature of propagation, these waves are get blocked at some point by curvature of earth as shown below:


17. Satellite Communication In this communication, frequency band 5.9 GHz to 6.4 GHz is used for uplinking and 3.7 GHz to 2 GHz is used for down linking.

Topic 2 Modulation

1. Modulation Modulation is the process of variation of some characteristics of a carrier wave in accordance with the instantaneous value of a modulating signal.
2. Need for Modulation It is due to the fact that low frequency signal

3. Types of modulations
(i) Amplitude modulation (ii) Frequency modulation
(iii) Phase modulation (iv) Pulse modulation
4. Amplitude Modulation In amplitude modulation, the amplitude of the carrier is varied in accordance with the information signal.



11. Types of pulse modulation
(i) PAM (Pulse Amplitude Modulation) (ii) PDM (Pulse Duration Modulation)
(iii) PPM (Pulse Position Modulation) (iv) PCM (Pulse Code Modulation)
12. Internet It is a network of computers, printers disk drives or other devices, connected in a network topology that allows the device to communicate.
13. Local Area Network In is a group of computers and associated devices that share a common communication line or wireless link. Typically, connected device share the resources of a single processor or server within a small geographic area.
14. Wide Area Network A Wide Area Network (WAN) is a network that covers a broad area (i. e. any tele-communications network that links across metropolitan, regional, national or international boundaries) using leased telecommunication lines.
15. Client Computer Every computer that extracts information from a server is called a client computer.
16. Webpage A hypertext document connected to the world wide area is known as webpage. It may contain text, videos, etc.
17. Website A location connected to the internet that maintains one or more web pages.
18. Internet Service Providers An Internet Service Provider (ISP) is an organisation that provide services for accessing using or participating in the internet.
19. People use internet for many purposes like searching and viewing information on any topic of interest for sending electronic mails, for e-banking, e-shopping, e-booking, etc.
20. Electronic mail Electronic mail is the exchange of computer-stored messages by telecommunication.
21. Mobile telephony is the provision of telephone services to phones which may move around freely rather than stay fixed in one location. Mobile phones connect to a terrestrial cellular network of base stations, whereas satellite phones connect to orbiting satellites.
22. A cellular network or mobile network is a wireless network distributed over land areas called class, each served by at least one fixed location transceiver, known as a cell site or base station.
23. All network related works including handling of all the incoming and outgoing calls are managed by a central control room called Mobile. Telephone Switching Office (MTSO)
24. A telephone numbering plan is a type of numbering schemes used in telecommunication to assign telephone numbers to subscriber telephones or other telephony endpoints.
25. 1G is the first generation of mobile network which are based on analog radio signal.
2 G is based on narrow band digital signal. 3 G is the increased data transfer speed.
4 G is provide a high-speed internet facility.
26. Global positioning system is a space based satellite navigation system that provides location and time information in all weather conditions.
27. Twelve number of satellites is required for correct and accurate location indentification in the global positioning system.

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Units and Measurement Class 11 Notes Physics Chapter 2

1. Measurement
   The process of measurement is basically a comparison process. To measure a physical quantity, we have to find out how many times a standard amount of that physical quantity is present in the quantity being measured. The number thus obtained is known as the magnitude and the standard chosen is called the unit of the physical quantity.
2.  Unit
   The unit of a physical quantity is an arbitrarily chosen standard which is widely accepted by the society and in terms of which other quantities of similar nature may be measured.
3. Standard
   The actual physical embodiment of the unit of a physical quantity is known as a standard of that physical quantity.
   • To express any measurement made we need the numerical value (n) and the unit (μ). Measurement of physical quantity = Numerical value x Unit
   For example: Length of a rod = 8 m
   where 8 is numerical value and m (metre) is unit of length.
4. Fundamental Physical Quantity/Units
   It is an elementary physical quantity, which does not require any other physical quantity to express it. It means it cannot be resolved further in terms of any other physical quantity. It is also known as basic physical quantity.
   The units of fundamental physical quantities are called fundamental units.
   For example, in M. K. S. system, Mass, Length and Time expressed in kilogram, metre and second respectively are fundamental units.
5. Derived Physical Quantity/Units
   All those physical quantities, which can be derived from the combination of two or more fundamental quantities or can be expressed in terms of basic physical quantities, are called derived physical quantities.
   The units of all other physical quantities, which car. be obtained from fundamental units, are called derived units. For example, units of velocity, density and force are m/s, kg/m3, kg m/s2 respectively and they are examples of derived units.
6. Systems of Units
   Earlier three different units systems were used in different countries. These were CGS, FPS and MKS systems. Now-a-days internationally SI system of units is followed. In SI unit system, seven quantities are taken as the base quantities.
   (i) CGS System. Centimetre, Gram and Second are used to express length, mass and time respectively.
   (ii) FPS System. Foot, pound and second are used to express length, mass and time respectively.
   (iii) MKS System. Length is expressed in metre, mass is expressed in kilogram and time is expressed in second. Metre, kilogram and second are used to express length, mass and time respectively.
   (iv) SI Units. Length, mass, time, electric current, thermodynamic temperature, Amount of substance and luminous intensity are expressed in metre, kilogram, second, ampere, kelvin, mole and candela respectively.
7. Definitions of Fundamental Units
   
8. Supplementary Units
   Besides the above mentioned seven units,there are two supplementary base units. these are (i) radian (rad) for angle, and (ii) steradian (sr) for solid angle.
   
9. Advantages of SI Unit System
   SI Unit System has following advantages over the other Besides the above mentioned seven units, there are two supplementary base units. These are systems of units:
   
   (i) It is internationally accepted,
   (ii) It is a rational unit system,
   (iii) It is a coherent unit system,
   (iv) It is a metric system,
   (v) It is closely related to CGS and MKS systems of units,
   (vi) Uses decimal system, hence is more user friendly.
10. Other Important Units of Length
    For measuring large distances e.g., distances of planets and stars etc., some bigger units of length such as ‘astronomical unit’, ‘light year’, parsec’ etc. are used.
    • The average separation between the Earth and the sun is called one astronomical unit.
    1 AU = 1.496 x 10¹¹ m.
    • The distance travelled by light in vacuum in one year is called light year.
    1 light year = 9.46 x 10¹⁵ m.
    • The distance at which an arc of length of one astronomical unit subtends an angle of one second at a point is called parsec.
    1 parsec = 3.08 x 10¹⁶ m
    • Size of a tiny nucleus = 1 fermi = If = 10^-15 m
    • Size of a tiny atom = 1 angstrom = 1A = 10^-10 m
11. Parallax Method
    This method is used to measure the distance of planets and stars from earth.
    Parallax. Hold a pen in front of your eyes and look at the pen by closing the right eye and ‘ then the left eye. What do you observe? The position of the pen changes with respect to the background. This relative shift in the position of the pen (object) w.r.t. background is called parallax.
    If a distant object e.g., a planet or a star subtends parallax angle 0 on an arc of radius b (known as basis) on Earth, then distance of that distant object from the basis is given by
    
    • To estimate size of atoms we can use electron microscope and tunneling microscopy technique. Rutherford’s a-particle scattering experiment enables us to estimate size of nuclei of different elements.
    • Pendulum clocks, mechanical watches (in which vibrations of a balance wheel are used) and quartz watches are commonly used to measure time. Cesium atomic clocks can be used to measure time with an accuracy of 1 part in 10¹³ (or to a maximum discrepancy of 3 ps in a year).
    • The SI unit of mass is kilogram. While dealing with atoms/ molecules and subatomic particles we define a unit known as “unified atomic mass unit” (1 u), where 1 u = 1.66 x 10^-27 kg.
    
12. Estimation of Molecular Size of Oleic Acid
    For this 1 cm³ of oleic acid is dissolved in alcohol to make a solution of 20 cm³. Then 1 cm³ of this solution is taken and diluted to 20 cm³, using alcohol. So, the concentration of the solution is as follows:
    
    After that some lycopodium powder is lightly sprinkled on the surface of water in a large trough and one drop of this solution is put in water. The oleic acid drop spreads into a thin, large and roughly circular film of molecular thickness on water surface. Then, the diameter of the thin film is quickly measured to get its area A. Suppose n drops were put in the water. Initially, the approximate volume of each drop is determined (V cm³).
    Volume of n drops of solution = nV cm³
    Amount of oleic acid in this solution
    
    The solution of oleic acid spreads very fast on the surface of water and forms a very thin layer of thickness t. If this spreads to form a film of area A cm², then thickness of the film
    
    If we assume that the film has mono-molecular thickness, this becomes the size or diameter of a molecule of oleic acid. The value of this thickness comes out to be of the order of 10^-9 m.
13. Dimensions
    The dimensions of a physical quantity are the powers to which the fundamental units of mass, length and time must be raised to represent the given physical quantity.
14. Dimensional Formula
    The dimensional formula of a physical quantity is an expression telling us how and which of the fundamental quantities enter into the unit of that quantity.
    It is customary to express the fundamental quantities by a capital letter, e.g., length (L), mass (AT), time (T), electric current (I), temperature (K) and luminous intensity (C). We write appropriate powers of these capital letters within square brackets to get the dimensional formula of any given physical quantity.
15. Applications of Dimensions
    The concept of dimensions and dimensional formulae are put to the following uses:
    (i) Checking the results obtained
    (ii) Conversion from one system of units to another
    (iii) Deriving relationships between physical quantities
    (iv) Scaling and studying of models.
    The underlying principle for these uses is the principle of homogeneity of dimensions. According to this principle, the ‘net’ dimensions of the various physical quantities on both sides of a permissible physical relation must be the same; also only dimensionally similar quantities can be added to or subtracted from each other.
    
16. Limitations of Dimensional Analysis
    The method of dimensions has the following limitations:
    (i) by this method the value of dimensionless constant cannot be calculated.
    (ii) by this method the equation containing trigonometric, exponential and logarithmic terms cannot be analyzed.
    (iii) if a physical quantity in mechanics depends on more than three factors, then relation among them cannot be established because we can have only three equations by equalizing the powers of M, L and T.
    (iv) it doesn’t tell whether the quantity is vector or scalar.
17. Significant Figures
    The significant figures are a measure of accuracy of a particular measurement of a physical quantity.
    Significant figures in a measurement are those digits in a physical quantity that are known reliably plus the first digit which is uncertain.
18. The Rules for Determining the Number of Significant Figures
    (i) All non-zero digits are significant.
    (ii) All zeroes between non-zero digits are significant.
    (iii) All zeroes to the right of the last non-zero digit are not significant in numbers without decimal point.
    (iv) All zeroes to the right of a decimal point and to the left of a non-zero digit are not significant.
    (v) All zeroes to the right of a decimal point and to the right of a non-zero digit are significant.
    (vi) In addition and subtraction, we should retain the least decimal place among the values operated, in the result.
    (vii) In multiplication and division, we should express the result with the least number of significant figures as associated with the least precise number in operation.
    (viii) If scientific notation is not used:
    (a) For a number greater than 1, without any decimal, the trailing zeroes are not significant.
    (b) For a number with a decimal, the trailing zeros are significant.
19.  Error
    The measured value of the physical quantity is usually different from its true value. The result of every measurement by any measuring instrument is an approximate number, which contains some uncertainty. This uncertainty is called error. Every calculated quantity, which is based on measured values, also has an error.
20. Causes of Errors in Measurement
    Following are the causes of errors in measurement:
    
    Least Count Error. The least count error is the error associated with the resolution of the instrument. Least count may not be sufficiently small. The maximum possible error is equal to the least count.
    Instrumental Error. This is due to faulty calibration or change in conditions (e.g., thermal expansion of a measuring scale). An instrument may also have a zero error. A correction has to be applied.
    Random Error. This is also called chance error. It makes to give different results for same measurements taken repeatedly. These errors are assumed to follow the Gaussian law of normal distribution.
    Accidental Error. This error gives too high or too low results. Measurements involving this error are not included in calculations.
    Systematic Error. The systematic errors are those errors that tend to be in one direction, either positive or negative. Errors due to air buoyancy in weighing and radiation loss in calorimetry are systematic errors. They can be eliminated by manipulation. Some of the sources of systematic errors are:
    (i) intrumental error
    (ii) imperfection in experimental technique or procedure
    (iii) personal errors
21. Absolute Error, Relative Error and Percentage Error

1. Combination of Errors
   
2. IMPORTANT TABLES
   
   
   
   
   

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Physical World Class 11 Notes Physics Chapter 1

1. Science
   Science is a systematic and organised attempt to acquire knowledge about the surroundings through observations, experiments and verifications.
2. Scientific Method
   Several inter-related steps are involved in scientific method. Some of the most significant steps are as follows:
   • The systematic observations
   • Reasoning
   • Mathematical modelling
   • Theoretical prediction
3. Physics
   Physics is a fundamental science concerned with understanding the natural phenomena that occur in our universe.
   It has many branches such as Mechanics, Electromagnetism, Thermodynamics, Modem Physics, etc. Between 1600 and 1900, three broad areas were developed, which is together called Classical Physics. These three areas of study are classical mechanics, thermodynamics and electromagnetism. But by 1905 it became apparent that classical ideas failed to explain several phenomena. Then some new theories were developed in what is called Modem Physics such as Special Relativity, Quantum Mechanics, etc.
4. Scope and Excitement of Physics
   The scope of Physics is very broad and covers a wide range of magnitude of physical quantities such as length, mass, time, energy, etc.
   It deals with the macroscopic world like galaxies and universe as well as microscopic world like nucleus of an atom and fundamental particles like electrons, protons, neutrons etc.
   Immense excitement is involved in the study of physics since it explains every naturally occuring phenomena with a set of rules, so that clear understanding can be achieved. The challenge to carry out imaginative new experiments to unlock the secrets of nature, to verify or refute theories, is really exciting.
5. Physics in Relation to Other Sciences
   Physics is a very significant branch of science which plays a crucial role in understanding the developments pertaining to the other branches of science such as Chemistry, Biology etc.
   (i) Physics in relation to Mathematics. Study of physical variables led to the idea of differentiation, integration and differential equation. Meaningful interpretation of Mathematics becomes Physics.
   (ii) Physics in relation to Chemistry. The concept of X-ray diffraction and radioactivity has helped to distinguish between the various solids and to modify the periodic table.
   Understanding the bonding and the chemical structure of substances is easy with the help of the concept of interactions between various particles.
   (iii) Physics in relation to Astronomy. Optical telescopes of reflecting and refracting type enabled man to explore the space around. Discoveries like radio telescopes have revolutionised the study of Astronomy.
   (iv) Physics in relation to Biology. The conceptual study of pressure and its measurement has helped us to know blood pressure and hence the functioning of heart. Invention of X-rays developed the field of diagnosis. Electron and optical microscopic designs have revolutionised the study of medical science.
   (v) Physics in relation to Meteorology. The discoveries regarding the study of pressure variations help us to forecast the weather.
   Various other inventions of physics have opened new vistas of study in the field of sciences and social sciences.
6. Physics in Relation to Technology and Society
   Advancement in physics has led to new technologies and vice-versa. Sometimes technology gives rise to new dimension of physics; at other times physics generates new technology. In fact, the technological development is closely related to the application of science and physics in particular. Physics has a dominant influence on society. It has helped the human being to develop its ideas. Development of digital communication systems, rapid mass transport system, lasers making bloodless surgeries, etc., has made human life easy and pleasant.
   • There are four fundamental forces in nature that govern the diverse phenomena of the microscopic and macroscopic world. These are the ‘gravitational force’, the ‘electromagnetic force’; the ‘strong nuclear force’, and the ‘weak nuclear force’. Unification of forces is a basic quest in physics. The electromagnetic and the weak nuclear forces have now been unified and are seen as aspects of a single ‘electro-weak’ force. Attempts are being made to unify electro-weak and the strong force.
   • Conservation of energy, momentum, angular momentum, charge, etc., are considered to be the fundamental laws in physics. Conservation laws have a deep connection with symmetries of nature. Symmetries of space and time, and other types of symmetries play a central role in modem theories of fundamental forces in nature.
7. IMPORTANT TABLES
   Table 1.1 Some Physicists from Different Countries of the World and their Major Contributions

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Motion in a Straight Line Class 11 Notes Physics Chapter 3

• Introduction
Motion is one of the significant topics in physics. Everything in the universe moves. It might only be a small amount of movement and very-very slow, but movement does happen. Even if you appear to be standing still, the Earth is moving around the sun, and the sun is moving around our galaxy.
“An object is said to be in motion if its position changes with time”.
The concept of motion is a re’ live one and a body that may be in motion relative to one reference system, may be at rest relative to another.
There are two branches in physics that examine the motion of an object.
(i) Kinematics: It describes the motion of objects, without looking at the cause of the motion.
(ii) Dynamics: It relates the motion of objects to the forces which cause them.
• Point Object
If the length covered by the objects are very large in comparison to the size of the objects, the objects are considered point objects.
• Reference Systems
The motion of a particle is always described with respect to a reference system. A reference system is made by taking an arbitrary point as origin and imagining a co-ordinate system to be attached to it. This co-ordinate system chosen for a given problem constitutes the reference system for it. We generally choose a co-ordinate system attached to the earth as the reference system for most of the problems.
• Total Path Length (Distance)
For a particle in motion the total length of the actual path traversed between initial and final positions of the particle is known as the ‘total path length’ or distance covered by it.
• Types of Motion
In order to completely describe the motion of an object, we need to specify its position. For this, we need to know the position co-ordinates. In some cases, three position co-ordinates are required, while in some cases two or one position co-ordinate is required.
Based on these, motion can be classified as:
(i) One dimensional motion. A particle moving along a straight-line or a path is said to undergo one dimensional motion. For example, motion of a train along a straight line, freely falling body under gravity etc.
(ii) Two dimensional motion. A particle moving in a plane is said to undergo two dimensional motion. For example, motion of a shell fired by a gun, carrom board coins etc.
(iii) Three dimensional motion. A particle moving in space is said to undergo three dimensional motion. For example, motion of a kite in sky, motion of aeroplane etc.
• Displacement
Displacement of a particle in a given time is defined as the change in the position of particle in a particular direction during that time. It is given by a vector drawn from its initial position to its final position.
• Factors Distinguishing Displacement from Distance
—> Displacement has direction. Distance does not have direction.
—> The magnitude of displacement can be both positive and negative.
—> Distance is always positive. It never decreases with time.
—> Distance ≥ | Displacement |

• Uniform Speed and Uniform Velocity
Uniform Speed. An object is said to move with uniform speed if it covers equal distances in equal intervals of time, howsoever small these intervals of time may be.
Uniform Velocity. An object is said to move with uniform velocity if it covers equal displacements in equal intervals of time, howsoever small these intervals of time may be.
• Variable Speed and Variable Velocity
Variable Speed. An object is said to move with variable speed if it covers unequal distances in equal intervals of time, howsoever small these intervals of time may be.
Variable Velocity. An object is said to move with variable velocity if it covers unequal displacements in equal intervals of time, howsoever small these intervals of time may be.
• Average Speed and Average Velocity
Average Speed. It is the ratio of total path length traversed and the corresponding time interval.
Or

The average speed of an object is greater than or equal to the magnitude of the average velocity over a given time interval.
• Instantaneous Speed and Instantaneous Velocity
Instantaneous Speed. The speed of an object at an instant of time is called instantaneous speed.
Or
“Instantaneous speed is the limit of the average speed as the time interval becomes infinitesimally small”.

Instantaneous velocity
The instantaneous velocity of a particle is the velocity at any instant of time or at any point of its path.
or
“Instantaneous velocity or simply velocity is defined as the limit of the average velocity as the time interval Δt becomes infinitesimally small.”

• Acceleration
The rate at which velocity changes is called acceleration.

• Uniform Acceleration
If an object undergoes equal changes in velocity in equal time intervals it is called uniform acceleration.
• Average and Instantaneous Acceleration
Average Accelerating. It is the change in the velocity divided by the time-interval during which the change occurs.

Instantaneous Acceleration. It is defined as the limit of the average acceleration as the time-interval Δt goes to zero.

• Kinematical Graphs
The ‘displacement-time’ and the ‘velocity-time’ graphs of a particle are often used to provide us with a visual representation of the motion of a particle. The ‘shape’ of the graphs depends on the initial ‘co-ordinates’ and the ‘nature’ of the acceleration of the particle (Fig.)

The following general results are always valid
(i) The slope of the displacement-time graph at any instant gives the speed of the particle at that instant.
(ii) The slope of the velocity-time graph at any instant gives the magnitude of the acceleration of the particle at that instant.
(iii) The area enclosed by the velocity-time graph, the time-axis and the two co-ordinates at ,time instants t₁ to t₂ gives the distance moved by the particle in the time-interval from t₁ to t₂.
• Equations of Motion for Uniformly Accelerated Motion
For uniformly accelerated motion, some simple equations can be derived that relate displacement (x), time taken (f), initial velocity (u), final velocity (v) and acceleration (a). Following equation gives a relation between final and initial velocities v and u of an object moving with uniform acceleration a: v = u + at


• Suppose a body is projected vertically upward from a point A with velocity u.

In some problems it is convenient to take the downward direction as positive, in such case all the measurements in downward direction are considered as positive i.e., acceleration will be +g. But sometimes we may need to take upward as positive and if such case acceleration will be -g.
• Relative Velocity
Relative velocity of an object A with respect to another object B is the time rate at which the object A changes its position with respect to the object B.

—> The relative velocity of two objects moving in the same direction is the difference of the speeds of the objects.
—> The relative velocity of two objects moving in opposite direction is the sum of the speeds of the objects.
• IMPORTANT TABLES

Class 11 Physics Notes

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Motion in a Plane Class 11 Notes Physics Chapter 4

• Motion in a plane is called as motion in two dimensions e.g., projectile motion, circular motion etc. For the analysis of such motion our reference will be made of an origin and two co-ordinate axes X and Y.
• Scalar and Vector Quantities
Scalar Quantities. The physical quantities which are completely specified by their magnitude or size alone are called scalar quantities.
Examples. Length, mass, density, speed, work, etc.
Vector Quantities. Vector quantities are those physical quantities which are characterised by both magnitude and direction.
Examples. Velocity, displacement, acceleration, force, momentum, torque etc.
• Characteristics of Vectors
Following are the characteristics of vectors:
(i) These possess both magnitude and direction.
(ii) These do not obey the ordinary laws of Algebra.
(iii) These change if either magnitude or direction or both change.
(iv) These are represented by bold-faced letters or letters having arrow over them.
• Unit Vector
A unit vector is a vector of unit magnitude and points in a particular direction. It is used to specify the direction only. Unit vector is represented by putting a cap (^) over the quantity.

• Equal Vectors

• Zero Vector

• Negative of a Vector

• Parallel Vectors

• Coplanar Vectors
Vectors are said to be coplanar if they lie in the same plane or they are parallel to the same plane, otherwise they are said to be non-coplanar vectors.
• Displacement Vector
The displacement vector is a vector which gives the position of a point with reference to a point other than the origin of the co-ordinate system.

• Parallelogram Law of Vector Addition
If two vectors, acting simultaneously at a point, can be represented both in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then the resultant is represented completely both in magnitude and direction by the diagonal of the parallelogram passing through that point.

• Triangle Law of Vector Addition
If two vectors are represented both in magnitude and direction by the two sides of a triangle taken in the same order, then the resultant of these vectors is represented both in magnitude and direction by the third side of the triangle taken in the opposite order.

• Polygon Law of Vector Addition
If a number of vectors are represented both in magnitude and direction by the sides of a polygon taken in the same order, then the resultant vector is represented both in magnitude and direction by the closing side of the polygon taken in the opposite order.

• Properties of Vector Addition
Vector addition has following properties:


• Resolution of Vectors
It is a process of splitting a single vector into two or more vectors in different directions which together produce the same effect as is produced by the single vector alone.
The vectors into which the given single vector is splitted are called component of vectors. In fact, the resolution of a vector is just opposite to composition of vectors.

If the components of a given vector are perpendicular to each other, then they are called rectangular components.
• Position Vector

• Multiplication of Vectors
(i) Scalar product (Dot product). Scalar product of two vectors is defined as the product of the magnitude of two vectors with cosine of smaller angle between them.

• Properties of Scalar Product


• Properties of Cross Product


• Lami’s Theorem
Lami’s theorem states, “If a particle under the simultaneous action of three forces is in equilibrium, then each force has a constant ratio with the sine of the angle between the other two forces.”


• Projectile Motion
The projectile is a general name given to an object that is given an initial inclined velocity and which subsequently follows a path determined by the gravitational force acting on it and by the frictional resistance of the air. The path followed by a projectile is called its trajectory.
Equation of projectile motion. The general case of projectile motion corresponds to that of an object that has been given an initial velocity u at some angle 8 above (or below) the horizontal. The horizontal and vertical displacements x and y are given by


• Angular Displacement
Angular displacement of the object moving around a circular path is defined as the angle traced out by the radius vector at the centre of the circular path in a given time.
θ (angle) = arc/radius
θ —> the magnitude of angular displacement. It is expressed in radians (rad).
• Angular Velocity
Angular velocity of an object in circular motion is defined as the time rate of change of its angular displacement.

• Angular Acceleration
Angular acceleration of an object in circular motion is defined as the time rate of change of its angular velocity.

• Uniform Circular Motion
When a body moves in a circular path with a constant speed, then the motion of the body is known as uniform circular motion.
The time taken by the object to complete one revolution on its circular path is called time period. For circular motion, the number of revolutions completed per unit time is known as the frequency (v). Unit of frequency is 1 Hertz (1 Hz). It is found that

• Centripetal Acceleration
To maintain a particle in its uniform circular motion a radially inward acceleration should be continuously maintained. It is known as the centripetal acceleration.

• IMPORTANT TABLES

Class 11 Physics Notes

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Work, Energy and Power Class 11 Notes Physics Chapter 6

• Work is said to be done when a force applied on the body displaces the body through a certain distance in the direction of applied force.
It is measured by the product of the force and the distance moved in the direction of the force, i.e., W = F-S
• If an object undergoes a displacement ‘S’ along a straight line while acted on a force F that makes an angle 0 with S as shown.
The work done W by the agent is the product of the component of force in the direction of displacement and the magnitude of displacement.

• If we plot a graph between force applied and the displacement, then work done can be obtained by finding the area under the F-s graph.
• If a spring is stretched or compressed by a small distance from its unstretched configuration, the spring will exert a force on the block given by
F = -kx, where x is compression or elongation in spring, k is a constant called spring constant whose value depends inversely on unstretched length and the nature of material of spring.
The negative sign indicates that the direction of the spring force is opposite to x, the displacement of the free-end.

• Energy
The energy of a body is its capacity to do work. Anything which is able to do work is said to possess energy. Energy is measured in the same unit as that of work, namely, Joule.
Mechanical energy is of two types: Kinetic energy and Potential energy.
• Kinetic Energy
The energy possessed by a body by virtue of its motion is known as its kinetic energy.
For an object of mass m and having a velocity v, the kinetic energy is given by:
K.E. or K = 1/2 mv ²
• Potential Energy
The energy possessed by a body by virtue of its position or condition is known as its potential energy.
There are two common forms of potential energy: gravitational and elastic.
—> Gravitational potential energy of a body is the energy possessed by the body by virtue of its position above the surface of the earth.
It is given by
(U)P.E. = mgh
where m —> mass of a body
g —> acceleration due to gravity on the surface of earth. h —> height through which the body is raised.
—> When an elastic body is displaced from its equilibrium position, work is needed to be done against the restoring elastic force. The work done is stored up in the body in the form of its elastic potential energy.
If an elastic spring is stretched (or compressed) by a distance Y from its equilibrium position, then its elastic potential energy is given by
U= 1/2 kx²
where, k —> force constant of given spring
• Work-Energy Theorem
According to work-energy theorem, the work done by a force on a body is equal to the change in kinetic energy of the body.

• The Law of Conservation of Energy
According to the law of conservation of energy, the total energy of an isolated system does not change. Energy may be transformed from one form to another but the total energy of an isolated system remains constant.
• Energy can neither be created, nor destroyed.
• Besides mechanical energy, the energy may manifest itself in many other forms. Some of these forms are: thermal energy, electrical energy, chemical energy, visual light energy, nuclear energy etc.
• Equivalence of Mass and Energy
According to Einstein, mass and energy are inter-convertible. That is, mass can be converted into energy and energy can be converted into mass.

• Collision
Collision is defined as an isolated event in which two or more colliding bodies exert relatively strong forces on each other for a relatively short time.
Collision between particles have been divided broadly into two types.
(i) Elastic collisions (ii) Inelastic collisions
• Elastic Collision
A collision between two particles or bodies is said to be elastic if both the linear momentum and the kinetic energy of the system remain conserved.
Example: Collisions between atomic particles, atoms, marble balls and billiard balls.
• Inelastic Collision
A collision is said to be inelastic if the linear momentum of the system remains conserved but its kinetic energy is not conserved.
Example: When we drop a ball of wet putty on to the floor then the collision between ball and floor is an inelastic collision.
• Collision is said to be one dimensional, if the colliding particles, move along the same straight line path both before as well as after the collision.
• In one dimensional elastic collision, the relative velocity of approach before collision is equal to. the relative velocity of separation after collision.

• Coefficient of Restitution or Coefficient of Resilience
Coefficient of restitution is defined as the ratio of relative velocity of separation after collision to the relative velocity of approach before collision.

• Elastic and Inelastic Collisions in Two Dimensions




• Non-conservative Forces
A force is said to be non-conservative if the work done in moving from one point to another depends upon the the path followed.
Let W, be the work done in moving from A to B following the path 1. W2 through the path 2 and W3 through the path 3. Fig. (i).

Examples of non-conservative forces are :
(i) Force of friction (ii) Viscus force
Low of conservation of energy holds goods for both conservative and non-conservative forces.

Class 11 Physics Notes

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Law of Motion Class 11 Notes Physics Chapter 5

• Dynamics is the branch of physics in which we study the motion of a body by taking into consideration the cause i.e., force which produces the motion.
• Force
Force is an external cause in the form of push or pull, which produces or tries to produce motion in a body at rest, or stops/tries to stop a moving body or changes/tries to change the direction of motion of the body.
• The inherent property, with which a body resists any change in its state of motion is called inertia. Heavier the body, the inertia is more and lighter the body, lesser the inertia.
• Law of inertia states that a body has the inability to change its state of rest or uniform motion (i.e., a motion with constant velocity) or direction of motion by itself.
• Newton’s Laws of Motion
Law 1. A body will remain at rest or continue to move with uniform velocity unless an external force is applied to it.
First law of motion is also referred to as the ‘Law of inertia’. It defines inertia, force and inertial frame of reference.
I here is always a need of ‘frame of reference’ to describe and understand the motion of particle, lhc simplest ‘frame of reference’ used are known as the inertial frames.
A frame of referent, e is known as an inertial frame it, within it, all accelerations of any particle are caused by the action of ‘real forces’ on that particle.
When we talk about accelerations produced by ‘fictitious’ or ‘pseudo’ forces, the frame of reference is a non-inertial one.
Law 2. When an external force is applied to a body of constant mass the force produces an acceleration, which is directly proportional to the force and inversely proportional to the mass of the body.


Law 3. “To every action there is equal and opposite reaction force”. When a body A exerts a force on another body B, B exerts an equal and opposite force on A.
• Linear Momentum
The linear momentum of a body is defined as the product of the mass of the body and its velocity.

• Impulse
Forces acting for short duration are called impulsive forces. Impulse is defined as the product of force and the small time interval for which it acts. It is given by

Impulse of a force is a vector quantity and its SI unit is 1 Nm.
— If force of an impulse is changing with time, then the impulse is measured by finding the area bound by force-time graph for that force.
— Impulse of a force for a given time is equal to the total change in momentum of the body during the given time. Thus, we have

• Law of Conservation of Momentum
The total momentum of an isolated system of particles is conserved.
In other words, when no external force is applied to the system, its total momentum remains constant.

• Recoiling of a gun, flight of rockets and jet planes are some simple applications of the law of conservation of linear momentum.

• Concurrent Forces and Equilibrium
“A group of forces which are acting at one point are called concurrent forces.”
Concurrent forces are said to be in equilibrium if there is no change in the position of rest or the state of uniform motion of the body on which these concurrent forces are acting.
For concurrent forces to be in equilibrium, their resultant force must be zero. In case of three concurrent forces acting in a plane, the body will be in equilibrium if these three forces may be completely represented by three sides of a triangle taken in order. If number of concurrent forces is more than three, then these forces must be represented by sides of a closed polygon in order for equilibrium.


• Commonly Used Forces
(i) Weight of a body. It is the force with which earth attracts a body towards its centre. If M is mass of body and g is acceleration due to gravity, weight of the body is Mg in vertically downward direction.
(ii) Normal Force. If two bodies are in contact a contact force arises, if the surface is smooth the direction of force is normal to the plane of contact. We call this force as Normal force.

Example. Let us consider a book resting on the table. It is acted upon by its weight in vertically downward direction and is at rest. It means there is another force acting on the block in opposite direction, which balances its weight. This force is provided by the table and we call it as normal force.
(iii) Tension in string. Suppose a block is hanging from a string. Weight of the block is acting vertically downward but it is not moving, hence its weight is balanced by a force due to string. This force is called ‘Tension in string’. Tension is a force in a stretched string. Its direction is taken along the string and away from the body under consideration.

• Simple Pulley
Consider two bodies of masses m₁ and m₂ tied at the ends of an in extensible string, which passes over a light and friction less pulley. Let m₁ > m₂. The heavier body will move downwards and the lighter will move upwards. Let a be the common acceleration of the system of two bodies, which is given by


• Apparent Weight and Actual Weight
— ‘Apparent weight’ of a body is equal to its ‘actual weight’ if the body is either in a state of rest or in a state of uniform motion.
— Apparent weight of a body for vertically upward accelerated motion is given as
Apparent weight =Actual weight + Ma = M (g + a)
— Apparent weight of a body for vertically downward accelerated motion is given as
Apparent weight = Actual weight Ma = M (g – a).
• Friction
The opposition to any relative motion between two surfaces in contact is referred to as friction. It arises because of the ‘inter meshing’ of the surface irregularities of the two surfaces in contact.
• Static and Dynamic (Kinetic) Friction
The frictional forces between two surfaces in contact (i) before and (ii) after a relative motion between them has started, are referred to as static and dynamic friction respectively. Static friction is always a little more than dynamic friction.
The magnitude of kinetic frictional force is also proportional to normal force.

• Limiting Frictional Force
This frictional force acts when body is about to move. This is the maximum frictional force that can exist at the contact surface. We calculate its value using laws of friction.
Laws of Friction:
(i) The magnitude of limiting frictional force is proportional to the normal force at the contact surface.

(ii) The magnitude of limiting frictional force is independent of area of contact between the surfaces.
• Coefficient of Friction
The coefficient of friction (μ) between two surfaces is the ratio of their limiting frictional force to the normal force between them, i.e.,

• Angle of Friction
It is the angle which the resultant of the force of limiting friction F and the normal reaction R makes with the direction of the normal reaction. If θ is the angle of friction, we have

• Angle of Repose
Angle of repose (α) is the angle of an inclined plane with the horizontal at which a body placed over it just begins to slide down without any acceleration. Angle of repose is given by α = tan-1 (μ)

• Motion on a Rough Inclined Plane
Suppose a motion up the plane takes place under the action of pull P acting parallel to the plane.
• Centripetal Force
Centripetal force is the force required to move a body uniformly in a circle. This force acts along the radius and towards the centre of the circle. It is given by

where, v is the linear velocity, r is the radius of circular path and ω is the angular velocity of the body.
• Centrifugal Force
Centrifugal force is a force that arises when a body is moving actually along a circular path, by virtue of tendency of the body to regain its natural straight line path.
The magnitude of centrifugal force is same as that of centripetal force.



• Motion in a Vertical Circle
The motion of a particle in a horizontal circle is different from the motion in vertical circle. In horizontal circle, the motion is not effected by the acceleration due to gravity (g) whereas in the motion of vertical circle, the motion is not effected by the acceleration due to gravity (g) whereas in the motion of vertical circle, the value of ‘g’ plays an important role, the motion in this case does not remain uniform. When the particle move up from its lowest position P, its speed continuously decreases till it reaches the highest point of its circular path. This is due to the work done against the force of gravity. When the particle moves down the circle, its speed would keep on increasing.

Let us consider a particle moving in a circular vertical path of radius V and centre o tide with a string. L be the instantaneous position of the particle such that

Here the following forces act on the particle of mass ‘m’.
(i) Its weight = mg (verticaly downwards).
(ii) The tension in the string T along LO.

We can take the horizontal direction at the lowest point ‘p’ as the position of zero gravitational potential energy. Now as per the principle of conservation of energy,

From this relation, we can calculate the tension in the string at the lowest point P, mid-way point and at the highest position of the moving particle.
Case (i) : At the lowest point P, θ = 0°




When the particle completes its motion along the vertical circle, it is referred to as “Looping the Loop” for this the minimum speed at the lowest position must be √5gr
• IMPORTANT TABLES

Class 11 Physics Notes

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Systems of Particles and Rotational Motion Class 11 Notes Physics Chapter 7

• A rigid body is a body with a perfectly definite and unchanging shape. The distances between all pairs of particles of such a body do not change.
• Centre of Mass
For a system of particles, the centre of mass is defined as that point where the entire mass of the system is imagined to be concentrated, for consideration of its translational motion.
If all the external forces acting on the body/system of bodies were to be applied at the centre of mass, the state of rest/ motion of the body/system of bodies shall remain unaffected.
• The centre of mass of a body or a system is its balancing point. The centre of mass of a two- particle system always lies on the line joining the two particles and is somewhere in between the particles.


• Motion of centre of Mass
The centre of mass of a system of particles moves as if the entire mass of the system were concentrated at the centre of mass and all the external forces were applied at that point. Velocity of centre of mass of a system of two particles, m₁ and m₂ with velocity v₁ and v₂ is given
by,

• If no external force acts on the body, then the centre of mass will have constant momentum. Its velocity is constant and acceleration is zero, i.e., MV_cm = constant.
• Vector Product or Cross Product of two vectors


• Torque
Torque is the moment of force. Torque acting on a particle is defined as the product of the magnitude of the force acting on the particle and the perpendicular distance of the application of force from the axis of rotation of the particle.


• Angular Momentum
The angular momentum (or moment of momentum) about an axis of rotation is a vector quantity, whose magnitude is equal to the product of the magnitude of momentum and the perpendicular distance of the line of action of momentum from the axis of rotation and its direction is perpendicular to the plane containing the momentum and the perpendicular distance.

• Axis of Rotation
A rigid body is said to be rotating if every point mass that makes it up, describes a circular path of a different radius but the same angular speed. The circular paths of all the point masses have a common centre. A line passing through this common centre is the axis of rotation.
• A rigid body is said to be in equilibrium if under the action of forces/torques, the body remains in its position of rest or of uniform motion.
For translational equilibrium, the vector sum of all the forces acting on a body must be zero. For rotational equilibrium, the vector sum of torques of all the forces acting on that body about the reference point must be zero. For complete equilibrium, both these conditions must be fulfilled.
• Couple
Two equal and opposite forces acting on a body but having different lines of action, form a couple. The net force due to a couple is zero, but they exert a torque and produce rotational motion.
• Moment of Inertia
The rotational inertia of a rigid body is referred to as its moment of inertia.
The moment of inertia of a body about an axis is defined as the sum of the products of the masses of the particles constituting the body and the square of their respective perpendicular distance from the axis.
It is given by .

• Radius of Gyration
The distance of a point in a body from the axis of rotation, at which if whole of the mass of the body were supposed to be concentrated, its moment of inertia about the axis of rotation would be the same as that determined by the actual distribution of mass of the body is called radius of gyration.
If we consider that the whole mass of the body is concentrated at a distance K from the axis of rotation, then moment of inertia I can be expressed as I = MK²

• Theorem of Parallel Axes
According to this theorem, the moment of inertia I of a body about any axis is equal to its moment of inertia about a parallel axis through centre of mass, I_cm, plus Ma² where M is the mass of the body and V is the perpendicular distance between the axes, i.e.,
I = I_cm + Ma²
• Theorem of Perpendicular Axes
According to this theorem, the moment of inertia I of the body about a perpendicular axis is equal to the sum of moments of inertia of the body about two axes at right angles to each other in the plane of the body and intersecting at a point where the perpendicular axis passes, i.e.,

• Rolling Motion
The combination of rotational motion and the translational motion of a rigid body is known as rolling motion.

• Law of Conservation of Angular Momentum
According to the law of conservation of angular momentum, if there is no external couple acting, the total angular momentum of a rigid body or a system of particles is conserved.

• IMPORTANT TABLES

Class 11 Physics Notes

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Mechanical Properties of Solids Class 11 Notes Physics Chapter 9

• Inter molecular Force
In a solid, atoms and molecules are arranged in such a way that each molecule is acted upon by the forces due to the neighbouring molecules. These forces are known as inter molecular forces.
• Elasticity
The property of the body to regain its original configuration (length, volume or shape) when the deforming forces are removed, is called elasticity.
• The change in the shape or size of a body when external forces act on it is determined by the forces between its atoms or molecules. These short range atomic forces are called elastic forces.
• Perfectly elastic body
A body which regains its original configuration immediately and completely after the removal of deforming force from it, is called perfectly elastic body. Quartz and phospher bronze are the examples of nearly perfectly elastic bodies.
• Plasticity
The inability of a body to return to its original size and shape even on removal of the deforming force is called plasticity and such a body is called a plastic body.
• Stress
Stress is defined as the ratio of the internal force F, produced when the substance is deformed, to the area A over which this force acts. In equilibrium, this force is equal in magnitude to the externally applied force. In other words,

• Stress is of two types:
(i) Normal stress: It is defined as the restoring force per unit area perpendicular to the surface of the body. Normal stress is of two types: tensile stress and compressive stress.
(ii) Tangential stress: When the elastic restoring force or deforming force acts parallel to the surface area, the stress is called tangential stress.
• Strain
It is defined as the ratio of the change in size or shape to the original size or shape. It has no dimensions, it is just a number.
Strain is of three types:
(i) Longitudinal strain: If the deforming force produces a change in length alone, the strain produced in the body is called longitudinal strain or tensile strain. It is given as:

(ii) Volumetric strain: If the deforming force produces a change in volume alone, the strain produced in the body is called volumetric strain. It is given as:

(iii) Shear strain: The angle tilt caused in the body due to tangential stress expressed is called shear strain. It is given as:

• The maximum stress to which the body can regain its original status on the removal of the deforming force is called elastic limit.
• Hooke’s Law
Hooke’s law states that, within elastic limits, the ratio of stress to the corresponding strain produced is a constant. This constant is called the modulus of elasticity. Thus

• Stress Strain Curve
Stress strain curves are useful to understand the tensile strength of a given material. The given figure shows a stress-strain curve of a given metal.

• The curve from O to A is linear. In this region Hooke’s Proportional limit law is obeyed.
• In the region from A to 6 stress and strain are not . proportional. Still, the body regains its original dimension, once the load is removed.
• Point B in the curve is yield point or elastic limit and the corresponding stress is known as yield strength of the material.
• The curve beyond B shows the region of plastic deformation.
• The point D on the curve shows the tensile strength of the material. Beyond this point, additional strain leads to fracture, in the given material.
• Young’s Modulus
For a solid, in the form of a wire or a thin rod, Young’s modulus of elasticity within elastic limit is defined as the ratio of longitudinal stress to longitudinal strain. It is given as:

• Bulk Modulus
Within elastic limit the bulk modulus is defined as the ratio of longitudinal stress and volumetric strain. It is given as:

– ve indicates that the volume variation and pressure variation always negate each other.
• Reciprocal of bulk modulus is commonly referred to as the “compressibility”. It is defined as the fractional change in volume per unit change in pressure.
• Shear Modulus or Modulus of Rigidity
It is defined as the ratio of the tangential stress to the shear strain.
Modulus of rigidity is given by

• Poisson’s Ratio
The ratio of change in diameter (ΔD) to the original diameter (D) is called lateral strain. The ratio of change in length (Δl) to the original length (l) is called longitudinal strain. The ratio of lateral strain to the longitudinal strain is called Poisson’s ratio.

• Elastic Fatigue
It is the property of an elastic body by virtue of which its behaviour becomes less elastic under the action of repeated alternating deforming forces.
• Relations between Elastic Moduli
For isotropic materials (i.e., materials having the same properties in all directions), only two of the three elastic constants are independent. For example, Young’s modulus can be expressed in terms of the bulk and shear moduli.

• Breaking Stress
The ultimate tensile strength of a material is the stress required to break a wire or a rod by pulling on it. The breaking stress of the material is the maximum stress which a material can withstand. Beyond this point breakage occurs.


Hence, the elastic potential energy of a wire (energy density) is equal to half the product of its stress and strain.
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Oscillations Class 11 Notes Physics Chapter 14

• Periodic Motion
Motions, processes or phenomena, which repeat themselves at regular intervals, are called periodic.
• Oscillatory Motion
The motion of a body is said to be oscillatory motion if it moves to and fro about a fixed point after regular intervals of time. The fixed point about which the body oscillates is called mean position or equilibrium position.
• Simple Harmonic Motion
Simple harmonic motion is a special type of periodic oscillatory motion in which
(i) The particle oscillates on a straight line
(ii) The acceleration of the particle is always directed towards a fixed point on the line.
(iii) The magnitude of acceleration is proportional to the displacement of the particle from the

• Characteristics of SHM
The displacement x in SHM at time t is given by
x = A sin (ωt+ Ф )
where the three constants A, ω and Ф characterize the SHM, i.e., they distinguish one SHM from another. A SHM can also be described by a cosine function as follows:
x = A cos (ωt + δ)
• The displacement of an oscillating particle at any instant is equal to the change in its position vector during that time. The maximum value of displacement in an oscillatory motion on either side of its mean position is called “displacement amplitude” or “simple amplitude”.
Thus, amplitude A = x max.
• The time taken by an oscillating particle to complete one full oscillation to and fro about its mean (equilibrium) position is called the “time period” of SHM. It is given by

• Frequency
The number of oscillations in one second is called frequency. It is expressed in sec-1 or Hertz. Frequency and time period are independent of amplitude.

• Phase
The quantity (ωt+ Ф) is called the phase of SHM at time t; it describes the state of motion at that instant. The quantity Ф is the phase at time f = 0 and is called the phase constant or initial phase or epoch of the SHM. The phase constant is the time-independent term in the cosine or sine function.

• The force responsible for maintaining the S.H.M. is called restoring force.
If the displacement (x) from the equilibrium position is small, the restoring force (F) acting on the body is given by
F = -kx
where k is a force constant.
• Energy in S.H.M.
When a body executes SHM, its energy changes between kinetic and potential, but the total energy is always constant. At any displacement x from the equilibrium position:


• Springs in Series
If two springs, having spring constant k₁ and k₂, are joined in series, the spring constant of the combination is given by

• Springs in Parallel
If two springs, having spring constants k₁ and k₂, are joined in parallel, the spring constant of the combination is given by
k = k₁ + k₂
• When one spring is attached to two masses m₁ and m₂, then

• Simple Pendulum
A simple pendulum is the most common example of bodies executing S.H.M. An ideal simple pendulum consists of a heavy point mass body suspended by a weightless in extensible and perfectly flexible string from a rigid support about which it is free to oscillate.
• The time period of simple pendulum of length ‘l’ is given by

The time period of a simple pendulum depends on
(i) length of the pendulum and
(ii) the acceleration due to gravity (g).
• A second’s pendulum is a pendulum whose time period is. 2s. At a place where g = 9.8 ms^-2, the length of a second’s pendulum is found to be 99.3 cm (= 1 m).
• If a liquid of density p oscillates in a vertical U-tube of uniform cross sectional area A, then the time period of oscillation is given by

• If a cylinder of mass m, length L, density of material p and uniform area of cross section A, oscillates vertically in a liquid of density o, then the time period of oscillation is given by
• Undamped and Damped Simple Harmonic Oscillations
Undamped Simple Harmonic oscillations: When a simple harmonic system oscillates with a constant amplitude which does not change with time, its oscillations are called undamped simple harmonic oscillations.
Damped Simple Harmonic oscillations: When a simple harmonic system oscillates with a decreasing amplitude with time, its oscillations are called damped simple harmonic oscillations.
The angular frequency of the damped oscillator is given by

• A system is said to execute free oscillations, if on being displaced or disturbed from its position of equilibrium, it oscillates itself without outside interference.
When a system is compelled to oscillate with a frequency other than its natural frequency, it is said to execute forced oscillations.
The external force which causes forced oscillation, is of sinusoidal nature. It is given as

• Resonance is the phenomenon of setting a body into oscillations with large amplitude under the influence of some external periodic force whose frequency is exactly equal to the natural frequency of the given body. Such oscillations are called the “resonant oscillations”.
• The two or more oscillations linked together in such a way that the exchange of energy takes place between them are called coupled oscillators. The oscillations produced by coupled oscillators are known as coupled oscillations.
• The speed of a mechanic wave depends upon the properties of the medium in which it is travelling. If E is the elastic constant and ρ is the density of the medium then the speed of the wave is given by

• In case of electric magnetic waves, we know that they are the combinations of the oscillation of electric and magnetic fields in perpendicular directions. Their speed of propagation depends upon the permitivity and the permeability of the medium. If μ₀ is permeability and ε₀ is the permitivity of the medium in vaccum, then

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Waves Class 11 Notes Physics Chapter 15

• Waves
Wave is a form of disturbance which travels through a material medium due to the repeated f periodic motion of the particles of the medium about their mean positions without any actual
transportation of matter.
• Characteristics of wave
The characteristics of waves are as follows:
(i) The particles of the medium traversed by a wave execute relatively small vibrations about their mean positions but the particles are not permanently displaced in the direction of propagation of the wave.
(ii) Each successive particle of the medium executes a motion quite similar to its predecessors along/perpendicular to the line of travel of the wave.
(iii) During wave motion only transfer of energy takes place but not that of a portion of the medium.
Waves are mainly of three types: (a) mechanical or elastic waves, (b) electromagnetic waves and (c) matter waves.
• Mechanical waves
Mechanical waves can be produced or propagated only in a material medium. These waves are governed by Newton’s laws of motion. For example, waves on water surface, waves on strings, sound waves etc.
• Electromagnetic Waves
These are the waves which require no material medium for their production and propagation, i.e., they can pass through vacuum and any other material medium. Common examples of electromagnetic
waves are visible light; ultra-violet light; radiowaves, microwaves etc.
• Matter waves
These waves are associated with moving particles of matter, like electrons, protons, neutrons etc. Mechanical waves are of two types:
(i) Transverse wave motion, (ii) Longitudinal wave motion,
• Transverse wave motion
In transverse waves the particles of the medium vibrate at right angles to the direction in which the wave propagates. Waves on strings, surface water waves and electromagnetic waves are transverse waves. In electromagnetic waves (which include light waves) the disturbance that travels is not a result of vibrations of particles but it is the oscillation of electric and magnetic fields which takes place at right angles to the direction in which the wave travels.
• Longitudinal wave motion
In these types of waves, particles of the medium vibrate to and fro about their mean position along the direction of propagation of energy. These are also called pressure waves. Sound waves are longitudinal mechanical waves.
• Wavelength
The distance travelled by the disturbance during the time of one vibration by a medium particle is called the wavelength (λ). In case of a transverse wave the wavelength may also be defined as the distance between two successive crests or troughs. In case of a longitudinal wave, the wavelength (λ) is equal to distance from centre of one compression (or refraction) to another.
• Wave Velocity
Wave velocity is the time rate of propagation of wave motion in the given medium. It is different from particle velocity. Wave velocity depends upon the nature of medium.
Wave velocity (υ) = frequency (v) x wavelength (λ)
• Amplitude
The amplitude of a wave is the maximum displacement of the particles of the medium from their mean position.
• Frequency
The number of vibrations made by a particle in one second is called Frequency. It is represented by v. Its unit is hertz (Hz) v =1/T
• Time Period
The time taken by a particle to complete one vibration is called time period.
T = 1/v, it is expressed in seconds.
• The velocity of transverse waves in a stretched string is given by

where T is the tension in the string and μ is the mass per unit length of the string, μ is also called linear mass density of the string. SI unit of μ is kg m-1.
• The velocity of the longitudinal wave in an elastic medium is given by

where E is the modulus of elasticity of the medium and ρ is the density of the medium. In case of solids, E is Young’s modulus of elasticity (Y), then

• Newton’s Formula for the velocity of sound in Air
According to Newton, when sound waves travel in air or in a gaseous media, the change is taking place isothermally and hence, it is found that

Speed of sound in air at STP conditions, calculated on the basis of Newton’s formula is 280 ms^-1. However, the experimentally determined values is 332 ms^-1.
According to Laplace, during propagation of sound waves, the change takes place under adiabatic conditions because gases are thermal insulators and compressions and refractions are alternately taking place with a high frequency.

• Factors Influencing Velocity of Sound
The velocity of sound in any gaseous medium is affected by a large number of factors like density, pressure, temperature, humidity, wind velocity etc.
(i) The velocity of sound in a gas is inversely proportional to the square root of density of the gas.
(ii) The velocity of sound is independent of the change in pressure of the gas, provided temperature remains constant.
(iii) The velocity of sound in a gas is directly proportional to the square root of its absolute temperature.
(iv) The velocity of sound in moist air is greater than the velocity of sound in dry air.
(v) If wind flows at an angle θ to the direction of propagation of sound, the velocity of sound is v + w cos θ, where w is the velocity of wind.
• General Equation of Progressive Waves
“A progressive wave is one which travels in a given direction with constant amplitude, i.e., without attenuation.”
As in wave motion, the displacement is a function of space as well as time, hence displacement relation is expressed as a combined function of position and time as:
y (x,t) = A sin (kx — ωt + Ф)
We may also choose a cosine function instead of sine function. Here A, K, ω and Ф are four constant for a given wave and are known as amplitude, angular wave number, angular frequency and initial phase angle of given wave.
• Relation between phase and path difference

• A wave motion can be reflected from a rigid as well as from a free boundary. A travelling wave, at a rigid boundary or a closed end, is reflected with a phase reversal but the reflection at an open boundary takes place without any phase change.
• The Principle of Superposition of Wave
When any number of waves meet simultaneously at a point in a medium, the net displacement at a given time is the algebraic sum of the displacements due to each wave at that time.

• Standing waves or Stationary waves
When two sets of progressive wave trains of the same type (i.e., both longitudinal or both transverse) having the same amplitude and time period/frequency/ wavelength travelling with same speed along the same straight line in opposite directions superimpose, a new set of waves are formed. These are called stationary waves or standing waves.

• Progressive Waves
1. The disturbance progresses on wards; it being handed over from particle to particle. Each particle executes the same type of vibration as the preceding one, though at a different time.
2. The waves are in the form of crests and troughs, i.e., sine/cosine functions, which move on wards with a definite velocity.
3. Every particle has the same amplitude; which it attains in its own time depending upon the progress of the wave.
4. The phase of every particle varies continuously from 0 to 2π .
5. No particle remains permanently at rest. Twice during each vibration, the particles are momentarily at rest. Different particles attain this position at different times.
6. All the particles have the same maximum velocity which they attain one after another, as the wave advances.
7. There is a regular flow of energy across every plane along the direction of propagation of the wave. The average energy in a wave is half potential and half kinetic.

• Stationary Waves
1. The disturbance is stationary, there being no forward or backward movement of the wave. Each particle has its own vibration characteristics.
2. The waves have the appearance of a sine/cosine function, which shrink to a straight line, twice in each vibration. It never advances.
3. Every particle has a fixed allotted amplitude. Some have zero amplitude (nodes) aiJ some have maximum amplitude (antinodes) always. Each partic1eattains this at the same given moment.
4. All the particles in one-half of the waves have a fixed phase and all the particles in the other half of the wave have the same phase in the opposite direction simultaneously.
5. There are particles -which are permanently at rest (nodes) and all other particles have their own allotted maximum displacement, which they attain simultaneously. These particles are momentarily at rest twice in each vibration, all at the same time.
6. All the particles attain their individual allotted velocities depending upon their positions, simultaneously. Two particles (nodes) in one wave form have zero velocities all the time.
7. There is no flow of energy at all, across any plane. Each particle has its own allotted individual energy. They all attain their values of RE. at one time and all energy becomes KB. at another given time.

• When a stationary wave is set up in a string of length l fixed at its two ends, in the simplest mode of vibration, nodes are formed at the fixed ends and an antinode is formed at the middle point. The frequency of fundamental mode of vibration (or first harmonic) is given by

• Frequency of the Stretched String
In general, if the string vibrates in P loops, the frequency of the string under that mode is given by

Based on this relation three laws of transverse vibrations of stretched strings arise. They are law of length, law of tension and law of mass.
• Law of Length
The fundamental frequency v is inversely proportional to,the length L of the stretched string.

• Law of Tension
The fundamental frequency is directly proportional to the square root of the tension in the string.

• Law of Mass
The fundamental frequency is inversely proportional to the square root of mass per unit length of the given string when L and T are kept constants.

• Beats
The phenomenon of regular rise and fall in the intensity of sound, when two waves of nearly equal frequencies travelling along the same line and in the same direction superimpose each other is called beats.
One rise and one fall in the intensity of sound constitutes one beat and the number of beats per second is called beat frequency. It is given as:
v_b = (v₁-v₂)
where v₁ and v₂ are the frequencies of the two interfering waves; v₁ being greater than v₂.
• Doppler Effect
According to Doppler’s effect, whenever there is a relative motion between a source of sound and listener, the apparent frequencies of sound heard by the listener is different from the actual frequency of sound emitted by the source.
For sound the observed frequency v’ is given by

Here v = true frequency of wave emitted by the source, v = speed of sound through the medium, v₀ the velocity of observer relative to the medium and vs the velocity of source relative to the medium. In using this formula, velocities in the direction OS (i.e., from observer towards the source) are treated as positive and those opposite to it are taken as negative.
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Thermodynamics Class 11 Notes Physics Chapter 12

• The branch of physics which deals with the study of transformation of heat into other forms of energy and vice-versa is called thermodynamics.
Thermodynamics is a macroscopic science. It deals with bulk systems and does not go into the
molecular constitution of matter.
• A collection of an extremely large number of atoms or molecules confined within certain boundaries
such that it has a certain values of pressure (P), volume (V) and temperature (T) is called a ; thermodynamic system.
• Thermal Equilibrium
A thermodynamic system is in an equilibrium state if the macroscopic variables such as pressure, volume, temperature, mass composition etc. that characterise the system do not change in time. In thermal equilibrium, the temperature of the two systems are equal.
• Zeroth Law of Thermodynamics
This law identifies thermal equilibrium and introduces temperature as a tool for identifying f equilibrium. According to this law “If two systems are in thermal equilibrium with a third system then those two systems themselves are in equilibrium.”
• Heat, Work and Internal Energy
— Energy that is transferred between a system and its surroundings whenever there is temperature difference between the system and its surroundings is called heat.
— Work is said to be done if a body or a system moves through a certain distance in the direction of the applied force. It is given as
dW = PdV
where P is the pressure of the gas in the cylinder.
— If we consider a bulk system consisting of a large number of molecules, then internal energy ; of the system is the sum of the kinetic energies and potential energies of these molecules.
This energy is possessed by a system due to its molecular motion and molecular configuration. The internal energy is denoted by U.
U = E_k + E_p
where E_k and E_p represent the kinetic and potential energies of the molecules of the system.
• Internal energy of a system is a macroscopic variable and it depends only on the state of the system. Its value depends only on the given state of the system and does not depend on the path taken to arrive that state.
• First Law of Thermodynamics
The first law of thermodynamics is simply the general law of conservation of energy applied to any system. According to this law, “the total heat energy change in any system is the sum of the internal energy change and the work done.”
When a certain quantity of heat dQ is subjected to a system, a part of it is used in increasing the internal energy by dU and a part is used in performing external work dW, hence
dQ = dU + dW
• For gases, the specific heat capacity depends on the process or the conditions under which heat capacity transfer takes place. There are mainly two principal specific heat capacities for a gas. These are specific heat capacity at constant volume and specific heat capacity at constant pressure.
• From First Law of Thermodynamics we find a relation between two principal specific heats of an ideal gas. According to the relation
C_p-C_υ = R
Here C_p and C_υ are molar specific heats under constant pressure and constant volume condition respectively.
The specific heat capacity of a gas at constant pressure is greater than the specific heat capacity of the gas at constant volume i.e. C_p > C_υ. Reason is that when heat supplied to a gas at constant volume, no work would be done by the gas against the external pressure and all the energy is used to raise the temperature of the gas. On the other hand when the heat is supplied to the gas at constant pressure, its volume increases and the heat energy supplied to it is used to increase the temperature of the gas as well as in doing the work against the external pressure.
The difference, between the two specific heats is the thermal equivalent of the work done in expanding the gas against the external pressure.
• Expression for the Relation between C_p and C_υ
Let P, V and T be the pressure, volume and absolute temperature initially of one mole of an ideal gas.
Case (i): The heat dQ is supplied to the gas at constant volume so that the temperature increases to T + dT.


• Thermodynamic State Variables
Thermodynamic state variables of a system are the parameters which describe equilibrium states of the system. For example, equilibrium state of gas is completely specified by the values of pressure, volume, temperature, mass and composition.
• Equation of State
The equation of state represents the connection between the state variables of a system. For example, the those equation of state of an ideal/perfect gas in represented as
PV = μRT
where g is number of moles of the gas and R is gas constant for one mole of the gas.
• Thermodynamic state variables are of two kinds, extensive and intensive. Extensive variables indicate the size of the system but intensive variables do not indicate the size. Volume, mass, internal energy of a system are extensive variables but pressure, temperature and density are intensive variables.
• Thermodynamic Processes
Any process in which the thermodynamic variables of a thermodynamic system change is known as thermodynamic process.
• Quasi-Static Processes
Processes that are sufficiently slow and do not involve accelerated motion of piston and/or large temperature gradient are quasi-static processes.
In this process, the change in pressure or change in volume or change in temperature of the system is very small.
• Isothermal Process
A change in pressure and volume of a gas without any change in its temperature, is called an isothermal change. In such a change, there is a free exchange of heat between the gas and its surroundings.
• Adiabatic Process
A process in which no exchange of heat energy takes place between the gas and the surroundings, is called an adiabatic process.
• The work done dW under isothermal change is given by

• P-V Diagram
A graph representing the variation of pressure with the variation of volume is called P-V diagram. The work done by the thermodynamic system is equal to the area under P-V diagram. It is given as

• Reversible Process
A process which can retrace so that the system passes through the same states is called a reversible process, otherwise it is irreversible.
Irreversibility arises mainly from two causes:
(i) Many processes like free expansion or an explosive chemical reaction take the system to non-equilibrium states.
(ii) Most processes involve friction, viscosity and other dissipative effects.
• Second Law of Thermodynamics
This principle which disallows certain phenomena consistent with the First law of thermodynamics is known as the second law of thermodynamics.
Following are the two statements of second law of thermodynamics.
Kelvin-Planck Statement: It is impossible to construct an engine, operating in a cycle, to extract
heat from hot body and convert it completely into work without leaving any change anywhere i.e., 100% conversion of heat into work is impossible.
Clausius Statement: It is impossible for a self acting machine, operating in a cycle, unaided by any external energy to transfer heat from a cold body to a hot body. In other words heat can not flow itself from a colder body to a hotter body.
• A heat engine is a device by which a system is made to undergo a cyclic process that results
in conversion of heat to work. Basically, a heat engine consists of: (i) a heat source, (ii) a heat sink, and (iii) a working substance.
• Carnot’s Engine. He proposed a hypothetical engine working on a cyclic/reversible process operating between two temperatures. Its efficiency is independent of the working substance and is given by, η=1-T₂/T₁ where T₁ is the temperature of source and T₂ is the temperature of sink.
• According to Carnot’s theorem: (a) working between two given temperatures T₁ and T₂ of the hot and cold reservoirs respectively, no engine can have efficiency more than that of Carnot’s engine, and (b) the efficiency of the Carnot engine is independent of the nature of the working substance.
• Refrigerator
The process of removing heat from bodies colder than their surroundings is called refrigeration and the device doing so is called refrigerator.
In the refrigerator, heat is absorbed at low temperature and rejected at higher temperature with the help of external mechanical work. Thus, a refrigerator is a heat engine working backward and hence it is also called heat pump.
Refrigerator works on the reverse process of Carnot engine. By the work done on the system, heat is extracted from low temperature sink T₂ and passed on to high temperature source T₁. The coefficient of performance is given by

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Accounting for Not-for-Profit Organisation –  CBSE Notes for Class 12 Accountancy

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Topic1: Concepts of Partnership and Partnership Deed
1. Partnership According to Section 4 of the Indian Partnership Act, 1932, ‘Partnership is defined as, ‘the relation between persons who have, agreed to share the profits of a business carried on by all or any of them acting for all’.
2. Nature of Partnership Partnership is a separate business entity from accounting point of view, but from a legal viewpoint, partnership firm is not a separate legal entity from its partners.
3. Features of Partnership
(i) Two or more persons (ii) Agreement
(iii) Business (iv) Mutual agency
(v) Sharing of profits and losses (vi) Liability of partnership
(vii) Management and control (viii) Registration
4. Partnership Deed It is a document which contains the terms and conditions of partnership agreement.
A firm should have a partnership deed because:
(i) It regulates the rights, duties and liabilities of the partners.
(ii) It avoids disputes in future by acting as a proof.
5. Accounting Rules Applicable in the Absence of Partnership Deed
(i) Sharing of profits and losses – Equally.
(ii) Interest on capital – Not allowed.
(iii) Remuneration or salary to the partners – Not allowed.
(iv) Interest on partner’s loan – Allowed @ 6% per annum.
(v) Interest on drawings – Not charged.
Topic 2: Computation of Appropriation Items and Charge Items
1. Interest on Capital Interest on capital is allowed to compensate partners for contributing capital to the firm. It is paid only if the same has been provided for in the partnership.

JOURNAL
(i) Interest on Capital A/c                              Dr
To Partners Capital/Current A/c
(ii) Profit and Loss Appropriation A/c       Dr
To Interest on Capital A/c
Different cases related to interest on capital:
(i) When partnership deed is silent-Not allowed
(ii) When partnership deed provides that interest on capital is to be allowed.
There are three possible conditions:
(a) In case of loss Not allowed.
(b) In case of sufficient profit Interest on capital is allowed in full.
(c) In case of insufficient profit Interest on capital is allowed proportionately to the extent of profits. In this case profits are distributed in capital ratio.
(iii) If interest on capital is to be provided as a charge, it is allowed in full irrespective of profit or losses.
NOTE (i) Interest on additional capital will be calculated for the period it remained with the firm, i.e. from the date of introduction of additional capital to the last day of accounting year.
If Opening Capital is not given, it can be calculated as:
Opening Capital = Closing Capital + Drawings – Profits – Additional Capital
(ii) Interest on capital is calculated on the opening balance of the capital for the full year.
2. Interest on Partner’s Drawings The amount withdrawn by partners in cash or in kind for their personal use in anticipation of profits, is termed as drawings. When the partnership deed is silent, no interest on drawings is charged. Interest on drawings is calculated with reference to time period for which money was withdrawn. Interest on drawings in different cases is calculated as follows:
(i) If a partner withdraws a fixed amount in the beginning, middle and end of each period.

Value of time under different circumstances will be as under:

(ii) When an unequal amount is withdrawn at different dates, the interest on drawings is calculated with the help of
(a) Simple method
(b) Product method

(iii) If the date of withdraw is not given, then the interest on total drawings for the year is calculated for a six month period on an average basis.
3. Accounting Treatment of Salary or Commission to a Partner
Salary or commission to a partner is to be allowed if the partnership agreement provides for the same. Salary or commission to a partner is an appropriation out of profits and not a charge against the profits, i.e. they are to be allowed only if there are profits and hence, must be transferred to the debit of profit and loss appropriation account and not to the debit of profit and loss account.
Commission may be allowed as a percentage of net profit before charging such commission or after charging such commission.

NOTE: Charges such as interest on partner’s loans, manager’s salary and commission must be deducted from profit before transferring it to profit and loss appropriation account.
4. Accounting Treatment of Interest on Partner’s Loan to the Firm Interest on partner’s loan is a charge against the profits and not an appropriation out of profits and hence, must be transferred to the debit of profit and loss account and not to the debit of profit and loss appropriation account.
NOTE: (i) If there is an agreement as to the rate of interest, partner is entitled to an interest on loan at an agreed rate of interest. If there is no agreement as to the rate of interest, partner is entitled to interest on loan @ 6% per annum.
(ii) Interest on partner’s loan is not recorded in the partner’s capital/current account but it should be recorded to the credit side of partner’s loan account if it is outstanding.
5. Rent Paid to a Partner It is a charge against the profit and not an appropriation out of profits. It is, therefore, debited to profit and loss account and credited to partner’s current account in case of fixed capitals or to partner’s capital account, when capitals are fluctuating.
Topic 3: Maintenance of Capital Accounts of Partners: Fixed and Fluctuating Capital
1. Fixed Capital Under this method, the capitals of the partners shall remain fixed, unless additional capital is introduced or a part of the capital is withdrawn, as per the agreement among the partners.
Two accounts are maintained under this method:
Capital accounts and current accounts.
Transactions related to introduction and withdrawal of capital are recorded in capital account, rest are recorded in current account.
Capital account will always have credit balance and current account may have credit or debit balance.

2. Fluctuating Capital Under this method, only one account, i.e. capital account of each partner is maintained.
All transactions are recorded in capital accounts. Generally, capital account has credit balance but in exceptional cases it may have debit balance due to heavy losses or withdrawals.

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Reconstitution of a Partnership Firm — Admission of a Partner –  CBSE Notes for Class 12 Accountancy

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Topic 1: introduction and New Profit Sharing Ratio and Sacrificing Ratio
1. Meaning Admission of a partner is one of the modes of reconstituting the firm under which old partnership comes to an end and a new one between all partners (including incoming partner) comes into existence.
According to Section 31(1) of Indian Partnership Act, 1932, ‘A new partner can be admitted only with the consent of all the existing partners’.
A new partner is admitted for the following purposes:
(i) For procuring additional capital.
(ii) For acquiring additional managerial skills.
A newly admitted partner acquires following two main rights in the firm:
(i) Share in the future profits of the firm.
(ii) Share in the assets of the firm.
2. Adjustments Required at the Time of Admission of a New Partner
(i) Calculation of new profit sharing ratio and sacrificing ratio.
(ii) Accounting treatment of goodwill.
(iii) Accounting treatment of revaluation of assets and re-assessment of liabilities.
(iv) Accounting treatment of reserves accumulated profit and losses.
(v) Adjustment of capital.
3. New Profit Sharing Ratio The ratio in which all the partners (including incoming partner) share the future profits and losses is known as new profit sharing ratio.
New Profit Sharing Ratio = Old Ratio – Sacrificing Ratio
4. Sacrificing Ratio It is the ratio in which the old partners have agreed to sacrifice their share of profits in favour of new or incoming partner.
Sacrificing Ratio = Old Ratio – New Ratio
Topic 2: Treatment of Goodwill
1. Accounting Treatment of Goodwill When a new partner is admitted, his share in future profits of the firm is equal to the sacrifice of profit by an existing partner or partners of the firm, the amount he pays to compensate this sacrifice is called goodwill.
2. Various cases related to the treatment of goodwill
(i) When premium for goodwill is paid privately by new partner.
In case a new partner pays premium to the old partners privately or directly or outside the business, it will not be recorded because it is an out of business transaction. However, entry will be passed for capital brought in by new partner.
(ii) When premium for goodwill is brought in business by new partner in cash and retained in the business
(a) Write-off the existing goodwill (if any) appearing in the books of the firms
Old Partner’s Capital A/c Dr [In old ratio]
To Goodwill A/c
(b) For bringing premium for goodwill and capital in cash:
Cash/Bank A/c Dr
To Premium for Goodwill A/c To New Partner’s Capital A/c
(c) For distributing premium to sacrificing (old) partners in their sacrificing ratio:
Premium for Goodwill A/c Dr
To Sacrificing Partner’s Capital A/c
(iii) When premium for goodwill is brought in kind
(a) For bringing premium for goodwill and capital in assets:
Assets A/c Dr
To Premium for Goodwill A/c To New Partner’s Capital A/c
(b) For distributing premium to sacrificing (old) partners in their sacrificing ratio:
Premium for Goodwill A/c Dr
To Sacrificing Partner’s Capital A/c
(iv) When premium for goodwill is brought in by new partner and is withdrawn by old (sacrificing) partners folly or partly.
(a) For bringing premium for goodwill in cash by new partner
Cash/Bank A/c Dr [Amount of premium]
To Premium for Goodwill A/c
(b) For sharing of premium for goodwill by sacrificing partners
Premium for Goodwill A/c Dr [Amount of premium]
To Sacrificing Partner’s Capital A/c (in sacrificing ratio)
(c) For withdrawal of premium money by sacrificing partners fully/partly
Sacrificing Partner’s Capital A/c (Amount withdrawn) Dr To Cash/Bank A/c
(v) When a new partner brings only a part of premium for goodwill in cash
(a) For amount brought in by incoming Partner
Cash/Bank A/c Dr
To Premium for Goodwill A/c To New Partner’s Capital A/c
(b) For distributing the total goodwill due from incoming partner to sacrificing (old) partners in their sacrificing ratio
Premium for Goodwill A/c Dr
New Partner’s Capital/ Current A/c Dr
To Sacrificing (old) Partner’s Capital/ Current A/c
(vi) When the new partner is unable to bring his share of premium for goodwill in cash or kind
New Partner’s Capital/Current A/c Dr
To Sacrificing (old) Partner’s Capital/Current A/c
3. Hidden Goodwill Hidden or inferred goodwill is the excess of desired total capital of the firm over the actual combined capital of all the partners. In case, the value of goodwill is not given at the time of admission of a new partner, it is required to be calculated on the basis of an inferred method of profit sharing ratio or capitalisation.

Topic 3: Revaluation of Assets and Re-assessment of Liabilities
1. Meaning of Revaluation Account The account which is prepared to record changes in the value of assets and liabilities at the time of admission, retirement, death and change in profit sharing ratio is called revaluation account.
2. Accounting Treatment
The relevant journal entries are:

Format of Revaluation Account

3. Accounting Treatment of Reserves, Undistributed Profits or Losses
The new partner is not entitled to any share in undistributed profits or losses appearing in the balance sheet at the time of admission, as these are earned by the old partners. So, these should be transferred to old partners’ capital/current account.
Journal entries passed will be
(i) For Undistributed Profits,
General Reserve A/c Profit and Loss A/c Workmen Compensation Fund A/c Investment Fluctuation Fund A/c between book value and market value]
To Old Partner’s Capital/Current A/c [Old ratio]
(ii) For Undistributed Losses
Old Partner’s Capital/Current A/c Dr [Old ratio]
To Profit and Loss A/c To Deferred Revenue Expenditure A/c
Topic 4: Adjustment of Capital
At the time of admission of a new partner, the partners may agree that their capitals should also be adjusted so as to be proportionate to their profit sharing ratio.
The capitals of partners may be adjusted in any of the following ways:
(i) Adjusting the capitals of old partners on the basis of the capital of incoming partner
(When the total capital of the new firm is not given)
Steps involved in adjusting the capitals of old partners:
Step 1 Calculate total capital of the firm on the basis of capital of new partner.
Step 2 Calculate the new capitals of each partner.
Step 3 Ascertain the present capital of old partners (adjusted).
Step 4 Calculate the surplus/deficit capital by comparing Step 2 and Step 3.
(ii) Determining the new partner’s capital on the basis of combined capital of old partners
Steps involved in the determination of capital of new partner:
Step 1 Calculate the adjusted old capitals of old partners (after all adjustments have been made.
Step 2 Calculate the total capital of new firm.
Step 3 Calculate the total capitals of new partners as follows:
Total Capital (Step 2) x Share of New Partner

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Accounting for Partnership: Basic Concepts –  CBSE Notes for Class 12 Accountancy

Any change in existing agreement of partnership amounts to reconstitution of a firm. As a result, the existing agreement comes to an end and a new agreement comes into existence and the firm continues.
1. Modes of Reconstitution of a Partnership Firm
Reconstitution of a firm can take place in any of the following ways
(i) Change in the profit sharing ratio of existing partners.
(ii) Admission of a new partner.
(iii) Retirement of an existing partner.
(iv) Death of a partner.
2. Change in Profit Sharing Ratio Among the Existing Partners
When one or more partners acquire an interest in the business from another partner(s), it is said to be a change in the profit sharing ratio in a partnership firm. A change in the profit sharing ratio among the existing partners means it is a reconstitution of the firm without admission, retirement or death of a new partner(s).
The sacrifice made or gain received by a partner is calculated by deducting the new share from the old share of a partner.
Sacrificing/(Gaining) Share = Old Share – New Share
Reconstitution of a Partnership Firm: Change in Profit Sharing Ratio 43
3. Adjustments Required at the Time of Change in Profit Sharing Ratio (i) Determination of Sacrificing Ratio and Gaining Ratio
New Profit Sharing Ratio It is the ratio in which the partners are to share profits/losses in future.
Sacrificing Ratio It is the ratio in which the partners have agreed to sacrifice their share of profit in favour of other partner or partners. This ratio is calculated by taking out the difference between old profit share and new profit share.
Sacrificing Ratio = Old Ratio – New Ratio
Gaining Ratio It is the ratio in which the partners have agreed to gain their share of profit from other partner(s). This ratio is calculated by taking out the difference between new profit share and old profit share.
Gaining Ratio = New Ratio – Old Ratio
(ii) Accounting Treatment of Goodwill
The entry to be passed for adjustment of goodwill, when there is a change in profit sharing ratio is
Gaining Partners’ Capital/Current A/c Dr [In gaining ratio]
To Sacrificing Partners’ Capital/Current A/c [In sacrificing ratio]
(Being the adjustment made for goodwill on change in profit sharing ratio)
Treatment of Existing Goodwill
Goodwill (if any) appearing in the books of the firm is written-off by debiting it to all partners’ capital accounts in their old profit sharing ratio and by crediting the goodwill account.
The entry is
All Partners’ Capital/Current A/c              Dr [In old ratio]
To Goodwill A/c                                      [With book value of goodwill]
(iii) Revaluation of Assets and Reassessment of Liabilities
(a) When Revised Values are to be Recorded in the Books of Accounts
An account titled ‘Revaluation account or ‘Profit and loss adjustment account is opened for revaluation of assets and reassessment of liabilities.
Format of Revaluation Account

(b) When Revised Values are not to be Recorded in the Books of Accounts
If partners decide to record the net effect of revaluation of assets and liabilities without affecting the old amount of assets and liabilities, a single adjusting entry ‘ involving the capital accounts of gaining partners and sacrificing partners is passed.
(iv) Accounting Treatment of Reserves, Accumulated Profits or Losses

Adjustment of Reserves and Accumulated Profits/Losses through Capital Accounts Only
A single adjusting entry involving the capital accounts of sacrificing and gaining partner is passed, when the partners decide to record net effect of reserves and accumulated profits/losses without affecting the old figures.
In Case of Profit
Gaining Partner’s Capital A/c                      Dr
To Sacrificing Partner’s Capital A/c
In Case of Loss
Sacrificing Partner’s Capital A/c                 Dr
To Gaining Partner’s Capital A/c

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Analysis of Financial Statements –  CBSE Notes for Class 12 Accountancy

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Topic 1: Introduction to Financial Statements Analysis
1. Financial Statement Analysis It is the systematic numerical representation of the relationship of one financial fact with the other to measure the profitability, operational efficiency, solvency and the growth potential of the business.
2. Types of Financial Statement Analysis
(i) External analysis
(ii) Internal analysis
(iii) Horizontal analysis
(iv) Vertical analysis
(v) Long-term analysis
(vi) Short-term analysis
3. Process of Financial Statement Analysis
(i) Rearrangement of data
(ii) Comparison
(iii) Analysis
(iv) Interpretation
4. Importance or Objectives of Financial Statement Analysis
(i) Judging the operational efficiency of the business.
(ii) Measuring the profitability.
(iii) Measuring short-term and long-term financial position.
(iv) Indicating the trend of achievements.
(v) Assessing the growth potential of the business.
(vi) Inter-firm comparison
5. Uses or Advantages of Financial Statement Analysis
(i) Security analysis
(ii) Credit analysis
(iii) Debt analysis
(iv) Dividend decision
(v) General business analysis
6. Limitations of Financial Statement Analysis
(i) Financial statement analysis ignore qualitative aspects like quality of management, labour force and public relations.
(ii) Suffering from the limitations of financial statements, which are as follows:
(a) Financial statements are historical in nature.
(b) Financial statements do not show price level changes hence, affect the analysis also.
(c) The results obtained by analysis of financial statements may be misleading due to window dressing.
(d) Financial statements are affected by the personal ability and bias of the analyst.
7. Parties Interested in Financial Statement Analysis and their Areas of Interest

Topic 2: Tools of Financial Statements Analysis
Tools of Financial Statements Analysis There are different tools of financial statements
analysis available to the analyst. The following tools are used to measure the operational efficiency and financial soundness of an enterprise.
The most common used techniques of financial analysis are:
1. Comparative financial statements
2. Common size statements
3. Ratio analysis
4. Cash flow statements
1. Comparative Financial Statements Statements used to compare the items of income statement i.e. profit and loss account and position statement i.e. balance sheet for ascertaining the trend of the performance and profitability of an enterprise are known as comparative financial statements.
(i) Comparative income statement It is a statement which shows in percentage term the total of income earned and expenses incurred during two or more accounting periods.
Format of Comparative Income Statement

(ii) Comparative balance sheet. It is a statement showing assets and liabilities of the business for two or more accounting periods. It also shows the percentage change in the monetary value of the assets and liabilities.
Format of Comparative Balance Sheet

2. Common Size Statement The statement wherein figures reported are converted into percentage to some common base is known as common size statement. Each percentage shows the relation of the individual item to its respective total.
(i) Common-size income statement The statement in which sales figure is assumed to be 100 and all other figures are expressed as a percentage of sales is known as common size income statement.
Format of Common Size Income Statement

(ii) Common-size balance sheet In common size balance sheet, the total of assets or liabilities is assumed to be 100 and figures are expressed as a percentage of the total.
Format of Common Size Balance Sheet

3. Ratio Analysis The mathematical expression that shows the relationships between various groups of items contained in the financial statements is known as ratio analysis.
4. Cash Flow Statement It shows the inflows and outflows of cash and cash equivalents of an enterprise by classifying cash flows into operating, investing and financing activities during a particular period and analysing the reasons for changes in balance of cash between the two balance sheets dates.

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Accounting for Share Capital –  CBSE Notes for Class 12 Accountancy

Topic 1: Introduction
1. Company A joint stock company is an artificial person, created by law, having separate entity distinct from its members with a perpetual succession and a common seal.
2. Characteristics or Features of a Company
(i) Artificial person (ii) Voluntary association (iii) Created by law
(iv) Capital divisible into transferable shares (v) Limited liability
(vi) Perpetual succession (vii) Common seal
(viii) Separate legal entity from its members (ix) May sue or be sued
3. Kinds of Companies
(i) Private companies According to Section 2 (68) of the Companies Act, 2013, it is a company with minimum paid-up share capital of ^ 1,00,000 or such higher amount as may be prescribed in the Companies Act, 2013 and which by its Articles of Association
(a) Restricts the right to transfer its shares, if any.
(b) Except in one person company, limits the number of its members excluding its present and past employee members to 200; if the past or present employee acquired the shares while in employment and continue to hold them. If any share is held jointly by two or more persons, they shall be treated as a single member.
(c) Prohibits any invitation to the public to subscribe for any securities of the company.
The minimum number of members required to form a private company is two. The name of a private company ends with the words, ‘Private Limited’.
(ii) Public company As per Section 2 (7) of Companies Act, 2013, public company is a company which
(a) is not a private company.
(b) has minimum capital of Rs 5 lakh or such higher paid-up capital as may be prescribed.
(c) is a private company, which is a subsidiary of a public company. Minimum requirement of a public company is seven persons.
(iii) One person company is a company which has only one person as a member. It is a company incorporated as a private company which has only one member. Rule 3 of the Companies (Incorporation) Rules, 2014 provides that:
(a) Only a natural person being an Indian citizen and resident in India can form one person company or can be nominee for the sole member of one person company.
(b) One person can form only one ‘one person company’ or become nominee of one such company.
(c) It cannot be formed for charitable purpose.
(d) It cannot carry out non-banking financial investment activities including investment in securities of any body corporate.
(e) Its paid-up share capital is not more than Rs 50 lakhs.
(f) Its average annual turnover should not exceed Rs 2 crores.
4. Share According to Section 2 (84) of the Companies Act, 2013, share means a share in the share capital of a company and includes stock. The capital of company is divided into a number of equal units. Each unit is called a share. A company may divide its capital into share of Rs 100, Rs 50, Rs 10, Rs 5 or even Rs 1 each.
5. Types of Shares
(i) Preference shares According to Section 43 (b) of the Companies Act, 2013, preference shares are the shares which carry the following two preferential rights:
(a) Preferential right of dividend to be paid as fixed amount or an amount calculated at a fixed rate, which may either be free of or subject to income tax.
(b) Return of capital on the winding up of the company before that of equity shares. Holders of preference shares are called preference shareholders.
(ii) Equity shares According to Section 43(a) of the Companies Act 2013, equity share is that share which is not a preference share. Equity shares are the most commonly issued class of shares which carry the maximum ‘risks and rewards’ of the business. The risks being losing part or all of the value of shares if the business incurs losses, the rewards being payment of higher dividends and appreciation in the market value.
6. Share Capital It is that part of the capital of a company, which is represented by the total nominal value of shares, which it has issued.
7. Kinds of Share Capital
(i) Authorised share capital According to Section 2(8) of Companies Act, 2013, ‘authorised capital’ or ‘nominal capital’ means such capital as is authorised by the memorandum of a company to be the maximum amount of share capital of a company.
(ii) Issued capital According to Section 2(50) of the Companies Act, 2013, issued capital means such capital as the company issues from time to time for subscription.
Subscribed capital According to Section 2(86) of the Companies Act, 2013, ‘subscribed capital’ means such part of the capital which is for the time being subscribed by the members of a company.
(a) Subscribed and fully paid-up Shares are said to be ‘subscribed and fully paid-up’ when the entire nominal (face) value is called and also paid-up by the shareholders.
(b) Subscribed but not fully paid-up Shares are said to be ‘subscribed but not fully paid-up’ when
• the company has called-up the entire nominal (face) value of the share but has not received it.
• the company has not called-up the entire nominal (face) value of share.
A reference has been made two terms
• Called-up capital According to Section 2(15) of the Companies Act, 2013, ‘called-up capital’ means such part of the capital, which has been called for payment. Thus, it means the amount of nominal (face) value called-up by the company to be paid by the shareholders towards the share capital.
• Paid-up share capital According to Section 2(64) of the Companies Act, 2013, ‘paid-up share capital’ or ‘share capital paid-up’ means the amount that the shareholder has paid and the company has received against the amount ‘called up’ against the shares towards share capital.
8. Reserve Capital It is that portion of uncalled share capital which shall not be capable of being called up except in the event and for the purpose of the company being wound up.
9. Capital Reserve ‘Capital reserve’ is the reserve which is not free for distribution as dividend. It is mandatory to create capital reserve in case of capital profits earned by the company. Reserves which are created out of capital profits are not readily available for distribution as dividend among the shareholders, e.g. premium on issue of shares of debentures, profits on re-issue of shares, profits prior to incorporation, premium on redemption of debentures.
10. Minimum Subscription It is the amount stated in the prospectus as the minimum amount that must be subscribed. Unless the sum payable on application for the sum so stated (minimum subscription) has been paid to and received by the company by cheque or other instruments, security cannot be allotted.
11. Presentation of Share Capital in Company’s Balance Sheet
As per Schedule III of Companies Act, 2013, share capital is to be disclosed in company’s balance sheet in the following manner


Topic 2: Accounting Treatment of Issue Shares
1. Terms of Issue of Shares
(i) Issue of shares at par When shares are issued at their face value, the shares are said to have been issued at par. i.e. issue price and face value are same.
(ii) Issue of shares at premium When shares are issued at a value that is higher than the face value of the shares, the shares are said to have been issued at premium, i.e. issue price is more than face value.
2. Utilisation of Securities Premium Reserve Section 52 (2) of the Companies Act, 2013 restrict the use of the amount received as premium on securities for the following purposes
(i) In purchasing its own shares (buy back) (Section 77A).
(ii) Issuing fully paid bonus shares to the members (Section 78).
(iii) Writing-off preliminary expenses of the company (Section 78).
(iv) Writing-off the expenses of, or the commission paid or discount allowed on any issue of securities or debentures of the company (Section 78).
(v) Providing for the premium payable on the redemption of any redeemable preference shares or of any debentures of the company (Section 78).
3. Accounting Treatment for Issue of Shares for Cash

4. Full Subscription of Shares When the number of shares applied for, is equal to the number of shares offered for subscription, the shares are said to be fully subscribed.
5. Over-Subscription of Shares When the number of shares applied for, is more than the number of shares offered for subscription, the shares are said to be oversubscribed. Allotment of shares cannot be made to all the applicants in full.
In case of over-subscription, following three alternatives are available (0 Rejection of applications (ii) Partial or pro-rata allotment (iii) Combination of pro-rata allotment and rejection

6. Under Subscription of Shares When the number of shares applied for, is less than the number of shares offered to the public, the shares are said to be under-subscribed.
7. Issue of Shares for Consideration other than Cash
I. Issue of Shares to Vendors
In this regard, the purchase of assets and issue of shares are to be treated as two separate transactions.
(i) (a) When Assets are Purchased
Assets A/c (Individually) Dr
To Vendor
(Being assets purchased from———-)
(b) When Business is Purchased
Sundry Assets A/c Dr
Goodwill A/c* Dr
To Sundry Liabilities A/c To Vendor
To Capital Reserve A/c*
(Being business purchased from vendor for purchase consideration of Rs——-)
NOTE ‘Vendor’ is credited with purchase consideration payable to him *Either goodwill or capital Reserve will come.
(ii) On Issue of Shares
(a) At Par Vendor
To Share Capital A/c
(b) At Premium Vendor
To Share Capital A/c To Securities Premium Reserve A/c
II. Issue of Shares to Promoters
Formation Expenses/ Incorporation Cost/Goodwill A/c Dr
To Share Capital A/c
(Being … share of Rs … each issued to promoters of the company)
III. Issue of Shares to Underwriters
(i) Making Underwriting Expenses Due
Underwriting Expenses A/c Dr
To Underwriters A/c (Being underwriting commission due)
(ii) Issuing Shares to Underwriters
Underwriters A/c Dr
To Share Capital A/c
(Being …shares issued @…….per share to underwriters)
8. Calls-in-arrears When one or more shareholders fail to pay their dues at the time of allotment or call, it is technically called calls-in-arrears.
Table F of the Companies Act, 2013, provides for the payment of interest on calls-in-arrears at a rate not exceeding 10% per annum.
9. Calls-in-advance The part of the whole amount received from the shareholders before the call is made, is called calls-in-advance.
This amount is shown on the liabilities side of the balance sheet as a separate item under the head ‘share capital’ but is not added to the amount of paid-up capital.
Table F of the Companies Act, 2013, provides for the payment of interest on calls-in-advance at a rate not exceeding 12% per annum.
10. Forfeiture of Shares
Forfeiture of shares means cancellation of shares and seizure of the amount already received from defaulting shareholders.
(i) Forfeiture of Shares Originally Issued at Par
Share Capital A/c Dr (Called-up money)
To Forfeited Shares A/c (Paid-up money)
To Share Unpaid Calls A/c (Unpaid money or calls-in-arrears)
(Being forfeiture of………….shares for non-payment of call of……….per share)
(ii) Forfeiture of Shares Originally Issued at Premium and Premium was Received
Share Capital A/c Dr (Called-up money)
To Forfeited Shares A/c (Paid-up money)
To Share Allotment A/c (Unpaid money excluding premium)
To Share Unpaid Call A/c (Unpaid money or calls-in-arrears)
(Being forfeiture of …… shares for non-payment
of allotment and call of…… per share)
(iii) Forfeiture of Shares Originally Issued at Premium and Premium was not Received
Share Capital A/c Dr (Called-up money)
Securities Premium Reserve A/c Dr (Unpaid premium)
To Forfeited Shares A/c (Paid-up money)
To Share Allotment A/c (Unpaid money including premium)
To Share Unpaid Call A/c (Unpaid money or calls-in-arrears)
(Being forfeiture of …… shares for non-payment
of allotment and call of ……per share)
11. Re-issue of Shares
The directors can either cancel or re-issue the forfeited shares. Shares forfeited can be re-issued at par, at premium or at a discount
In case, they are re-issued at par, accounting entry will be
Bank A/c Dr
To Share Capital A/c
In case, shares are re-issued at a discount, the amount of discount allowed on the re-issue of forfeited shares must not exceed the amount forfeited on re-issued shares. The discount allowed on re-issue of forfeited shares should be debited to the ‘share forfeiture account’. The journal entry will be
Bank A/c Dr [With the amount received on re-issue]
*Share Forfeiture A/c Dr [With the discount allowed on re-issue]
To Share Capital A/c [With the amount credited as paid-up]
*It is calculated as
Number of Shares Re-issued x (Paid-up Value – Re-issue Price Per Share)
If the forfeited shares are re-issued at a price higher than that paid-up, the excess is credited to securities premium reserve account. The journal entry will be
Bank A/c Dr
To Share Capital A/c To Securities Premium Reserve A/c
Transfer of Balance in Forfeited Share Account Forfeited Shares A/c Dr
To Capital Reserve A/c (Being balance of share forfeiture account transferred to capital reserve account)

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Financial Statements of a Company –  CBSE Notes for Class 12 Accountancy

1. Financial Statements The statements which are prepared to ascertain the profit earned or loss suffered and position of assets and liabilities at a particular date are known as financial statements. These are the final product of accounting process.
A set of financial statements as per Section 2(40) of the Companies Act, 2013 include (0 Balance sheet i.e. position statement
(ii) Statement of profit and loss i.e. income statement
(iii) Notes to accounts
(iv) Cash flow statement
Section 129 of the Companies Act, 2013 requires the company to prepare its financial statements every year in prescribed form i.e. Schedule III of the Companies Act, 2013.
2. Characteristics of Financial Statements
(i) Financial statements are historical documents as they relate to past period.
(ii) Financial statements are prepared in monetary terms.
(iii) Balance sheet reveals the financial position and statement of profit and loss shows the profitability of the business organisation.
3. Nature of Financial Statements
(i) Recorded facts
(ii) Accounting conventions
(iii) Postulates
4. Objectives of Financial Statements
(i) Financial statements provide the information about the earning capacity of the business.
(ii) Financial statements provide the information about the economic resources and obligation of an enterprise.
(iii) Financial statements also provide the information about the cash flows.
(iv) Financial statements supply the information useful for judging the management’s ability to utilise the resources of business effectively.
(v) Financial statements have to report the activities of the business organisation affecting the society, which is important in its social environment.
5. Essentials of Financial Statements
(i) Accurate information
(ii) Understandability
(iii) Comparable
(iv) Verifiable
(v) Relevant
(vi) Timeliness
6. Uses and Importance of Financial Statements
(i) Report on stewardship function
(ii) Basis for fiscal policies
(iii) Basis of granting of credit
(iv) Basis for prospective investors
(v) Guide to the value of the investment already made
(vi) Aids trade associations in helping their members
7. Limitations of Financial Statements
(i) Accounting concepts and conventions involve personal judgement, so these statements are not free from bias.
(ii) Qualitative aspects of financial statements are ignored.
(iii) The present value of assets and liabilities and price-level changes are ignored.
(iv) Financial statements are historical in nature and relate to past period only.
8. Users of Financial Statements
(i) Owners including shareholders and investors
(ii) Debentureholders and financial institutions (bankers)
(iii) Creditors
(iv) Management
(v) Employees
(vi) Government, tax authorities and regulators
9. Balance Sheet
It may be defined as a statement of assets and liabilities of the company, at a particular date. It must exhibit a true and fair view of the financial position at the close of the year. It is prepared and presented in the form prescribed in Schedule III Part I of the Companies Act, 2013, and is broadly divided into two parts, (i) Equity and liabilities (ii) Assets.
Proforma of Balance Sheet (As per Revised Schedule VI)

10. Statement of Profit and Loss The title of ‘profit and loss account’ is charged to statement of profit and loss. If shows the net result of business operations. Its form is prescribed in Schedule III, Part II of the Companies Act, 2013.

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Issue and Redemption of Debentures –  CBSE Notes for Class 12 Accountancy

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Topic 1: Introduction, Issue of Debentures and Various Cases from The Point of View of Redemption
1. Meaning of Debenture It means a document of companies indebtedness issued under the seal of the company and containing a contract for the repayment of the principal sum at a specified date with interest at a fixed rate.
2. Definition of Debenture According to Section 2(30) of the Companies Act, 2013 ‘Debenture includes debenture stock, bonds and any other instrument of the company evidencing a debt, whether constituting a charge on the assets of the company or not.’
3. Rond Bond is also an instrument of acknowledgement of debt. Bond is similar to debenture in terms of contents and texture. However, bonds can be issued without pre-determined rate of interest.
4. Issue of Debentures for Cash Debentures, like shares, issued for cash, may be issued
(i) at par (ii) at premium or (iii) at discount
(i) Issue of Debentures at Par Debentures are said to be issued at par when the issue price and face value of the debentures is same.
(ii) Issue of Debentures at a Premium Debentures are said to be issued at premium when the issue price is more than the face value.
(iii) Issue of Debentures at a Discount Debentures are said to be issued at discount when they are issued at a price below its nominal or face value.
5. Accounting Entries for Issue of Debentures for Cash

6. Issue of Debentures for Consideration other than Cash When debentures are issued to vendors against purchase of assets or against purchase of business, it is termed as issue of debentures for consideration other than cash. In this case, consideration for issue of debentures is not cash but the assets or business. Debentures in this instance can also be issued at par, premium or discount.
Following journal entries will be passed in this case:
(i) On Purchase of Assets or Business
(a) When assets are purchased
Sundry Assets A/c Dr
To Vendor’s A/c Dr
(b) When business is purchased
Sundry Assets A/c Dr
To Sundry Liabilities A/c To
Vendor’s A/c *To Capital Reserve A/c
(ii) On Issue of Debentures
(a) When debentures are issued at par
Vendor’s A/c Dr
To X% Debentures A/c
(b) When debentures are issued at premium
Vendor’s A/c Dr
To X% Debentures A/c To Securities Premium Reserve A/c
(c) When debentures are issued at discount
Vendor’s A/c Dr
Discount on Issue of Debentures A/c Dr
To X% Debentures A/c
(i) Purchase consideration is the amount paid by the purchasing company for purchasing of assets/business from another enterprise.
(ii) If the purchase consideration is greater than the value of the net assets acquired, the difference is debited to goodwill account.
(iii) If the amount of purchase consideration is lower than the value of net assets acquired, the difference is credited to capital reserve account.
(iv) Either of the two i.e. capital reserve or goodwill will come.

7. Issue of Debentures as Collateral Security When a company takes a loan, it may provide primary security on its assets. However, the lending institution may insist on some more assets as secondary or collateral security. In such a situation, the company may issue debentures to the lender as secondary or collateral security, such an issue of debentures is known as ‘debentures issued as collateral security’.
If the company fails to repay the loan along with the interest and the primary security is insufficient to repay the loan, only in that case the lender is free to use the debentures as collateral security. The lender may either present such debentures for redemption or sell them in the open market.
Debentures issued as collateral security can be dealt in two ways
(i) First Method (Without Passing Journal Entry) In this method, no journal entry is passed in the books for issue of debentures as collateral security.
However, the fact of debentures issued as collateral security is disclosed by way of information below debentures, which are shown as long-term borrowings under non-current liabilities or as short-term borrowings under current liabilities.
(ii) Second Method (With Journal Entry) Debentures issued as collateral security may be recorded in the books of accounts.
Following journal entry will be passed for issue of debentures as collateral security: Debenture Suspense A/c Dr [This appears on assets side]
To X% Debentures A/c [This appears on liabilities side]
When the loan is paid the above entry is cancelled by passing its reverse entry. In balance sheet, debentures issued as collateral security must be shown separately from other debentures.
Disclosure of Debenture Issued as Collateral Security in the Balance Sheet As debenture issued as collateral security is related to the loan of the company, therefore, it is shown in the note in which the loan secured by debenture is shown.

8. Interest on Debentures Interest on debentures is calculated at a fixed rate on its face value and is usually payable half yearly. Debenture interest is a charge on profit and it is not an appropriation of profit. Hence, interest on debentures is payable even if the company suffers a loss, i.e. does not earn profit.
According to Income Tax Act 1961, a company paying interest on debentures is required to deduct income tax at the prescribed rate from the gross amount of debenture interest before any payment is made to debenture holders, it is called Tax Deducted at Source (TDS).
Entries for interest on debentures are as follows
(i) When interest is due
Debentures Interest A/c Dr [With gross interest]
To Debentureholders’ A/c [With net interest]
To Income Tax Payable A/c [With income tax deducted]
(ii) When interest is paid
Debentureholders’ A/c Dr [With interest]
To Bank or Cash A/c
(iii) On payment of income tax to government Income Tax Payable A/c Dr
To Bank A/c
(iv) On transfer of interest on debentures to profit and loss account Statement of Profit and Loss Dr [With the amount of interest]
To Debenture Interest A/c

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