It is a current which produces in the region in which the electric field and hence the electric flux changes with time.

Displacement current, I_D = ε_o . dφ_E / dt

where, φ_E is the electric flux.

Ampere-Maxwell Law

where, μ_o = Permeability

= 4π * 10^-7 V / Am

Maxwell’s Equations

This equation is Ampere-Maxwell law.

Electromagnetic Waves

Electromagnetic waves are those waves in which electric and magnetic field vectors changes sinusoidally and are perpendicular to each other as well as at right angles to the direction of propagation of wave.

The equation of plane progressive electromagnetic wave can be written as E = E_o sin Ω (t – x / c) and B = B_o sin Ω (t – x / c). Where, Ω =2πv

Electromagnetic waves are produced by accelerated charge particles.

Properties of EM Waves

(i) These waves are transverse in nature.

(ii) These waves propagate through space with speed of light, i.e., 3 * 10⁸ m / s.

(iii) The speed of electromagnetic wave,

c = 1 / √μ_o ε_o

where, μ_oo is permittivity of free space,

∴ c = E_o / B_o

where E_o and B_o are maximum values of electric and magnetic field vectors.

[According to Maxwell, when a charged particle is accelerated, it produces electromagnetic wave. The total radiant flux at any instant is given by,

p = q²a² / 6 πε_oc²

(iv) The rate of flow of energy in an electromagnetic wave is described by the vector S called the poynting vector, which is ; defined by the expression,

S = 1 / μ_o E * B

SI unit of Sis watt/m².

(v) Its magnitude S is related to the rate at which energy is transported by a wave across a unit area at any instant.

(vi) The energy in electromagnetic waves is divided equally between electric field and magnetic field vectors.

(vii) The average electric energy density.

U_E = 1 / 2 ε_o E² = 1 / 4 ε_o E²_o

(viii) The average magnetic energy density,

U_B = 1 / 2 B² / μ_o = 1 / B²_o / μ_o

(ix) The electric vector is responsible for the optical effects of an electromagnetic wave.

(x) Intensity of electromagnetic wave is defined as energy crossing per unit area per unit time perpendicular to the directions of propagation of electromagnetic wave.

(xi) The intensity I is given by the relation,

I = < μ > c = 1 / 2 ε_o E²_oc

(xii) The existence of electromagnetic waves was confirmed by Hertz experimentally in 1888.

Propagation of Electromagnetic Waves

In radio wave communication between two places. the electromagnetic waves are radiated out by the transmitter antenna at one place which travel through the space and reach the receiving antenna at the other place.

Electromagnetic Spectrum

The arranged array of electromagnetic radiations in the sequence of their wavelength or frequency is called electromagnetic spectrum

Radio and microwaves are used in radio and TV communication,

Infrared rays are used to

(i) Treat muscular straw.
(ii) For taking photographs’ in fog or smoke.
(iii) In green house to keep plants warm.
(iv) In weather forecasting through infrared photography.

Ultraviolet rays are used

(i) In the study of molecular structure.
(ii) In sterilizing the surgical instruments.
(iii) In the detection of forged documents, £ringer prints.

X-rays are used

(i) In detecting faults, cracks, flaws and holes in metal products.
(ii) In the study of crystal structure.
(iii) For the detection of pearls in oysters.

γ – rays are used for the study of nuclear structure.

Earth’s Atmosphere

The gaseous envelope surrounding the earth is called earth’s atmosphere. It contain the following layers

(i) Troposphere This region extends upto a height of 12 km from earth’s surface.

(ii) Stratosphere This region extends from 12 km to 50 km. In this region, most of the atmospheric ozone is concentrated from 30 to 50 km. This layer is called ozone layer.

(iii) Mesosphere The region extends from 50 km to 80 km.

(iv) Ionosphere This region extends from 80 km to 400 km.

In ionosphere the electron density is very large in a region beyond 110 km from earth’s surface which extends vertically for a few kilometer.

This layer is called Kennelly Heaviside layer.

In ionosphere a layer having large electron density is found at height 250 km from earth’s surface, called Appleton layer.

There are four main layers in earth’s atmosphere having high density of electrons and positive ions, produced due to ionisation by the high energy particles coming from sun. star or cosmos. These layers play their effective role in space communication. These layers are D, E, F₁ and F₂.

(i) D-layer is at a virtual height of 65 km from surface of earth and having electron density = 10⁹ m^-3

(ii) E-layer is at a virtual height of 100 km, from the surface of earth, having electron density = 2 * 10¹¹ m^-3

(iii) F₁-layer is at a virtual height of 180 km from the surface of earth, having electron density = 3 * 10¹¹ m^-3

(iv) F₂ – layer is at a vertical height of about 300 km in night time and about 250 to 400 km in day time. The electron density of this layer is = 8 * 10¹¹ m^-3

Communication

Faithful transmission of information from one place to another place is called communication.

Optical fibers are used in optical communication.

Communication System

A communication system contains three main parts

(i) Transmitter It process and encode the information and make it suitable for transmission.

The message signal for communication can be analog signals or digital signals.

An analog signal can be converted suitably into a digital signal and vice-versa.

[An analog signal is that in which current or voltage value varies continuously with time.

A digital signal is a discontinuous function of time. Such a signal is usually in the form of pulses.]

(ii) Communication Channel The medium through which information propagate from transmitter to receiver 1S called communication channel.

(iii) Receiver It receives and decode the signal.

Analog Signal

A signal in which current or voltage changes its magnitude continuously with time, is called an analog signal.

Digital Signal

A signal in which current or voltage have only two values, is called a digital signal.

Note An analog signal can be converted suitable Into a digital signal and vice-versa.

Modulation

The process of superimposing the audio signal over a high frequency carrier wave is called modulation.

In the process of modulation anyone characteristic of carrier wave is varied in accordance with the instantaneous value of audio signal (modulating signal).

Need of Modulation

(i) Energy carried by low frequency audio waves (20 Hz to 20000 Hz) is very small.

(ii) For efficient radiation and reception of signal. the transmitting and receiving antennas should be very high approximately 5000 m.

(iii) The frequency range of audio signal is so small that overlapping of signals create a confusion.

Types of Modulation

(i) Amplitude Modulation In this type of modulation, the amplitude of high frequency carrier wave is varied in accordance to instantaneous amplitude of modulating signal.

Band width required for amplitude modulation

= twice the frequency of the modulating signal.

(ii) Frequency Modulation In this type of modulation, the frequency of high frequency carrier wave is varied in accordance to instantaneous frequency of modulating signal.

(iii) Pulse Modulation In this type of modulation, the continuous waveforms are sampled at regular intervals. Information is transmitted only at the sampling times.

Demodulation

The process of separating of audio signal from modulated signal is called demodulation.

Antenna

An antenna converts electrical energy into electromagnetic waves at transmitting end and pick up transmitted signal at receiving end and converts electromagnetic waves into electrical signal.

Modem

The term modem is contraction of the term modulator and demodulator. Modem is a device which can modulate as well as demodulate the signal. It connect one computer to another through ordinary telephone lines.

Fax (Facsimile Telegraphy)

The electronic reproduction of a document at a distant place is called FAX

Radio Waves

The radio waves are the electromagnetic waves of frequency ranging from 500 kHz to about 1000 MHz. These \V8VeS are used In the field of radio communication. With reference to the frequency range and wavelength range, the radio waves have been divided into various categories shown in table.

Frequency Range and Wavelength Range of Radio Waves

Propagation of Radio Waves

The three modes are discussed below.

(i) Ground Wave or Surface Wave Propagation It is suitable for low and medium frequency up to 2 MHz. It is used for local broad casting.

(ii) Sky Wave Propagation It is suitable for radio waves of frequency between 2 MHz to 30 MHz. It is used for long distance radio communication.

Critical Frequency The highest frequency of radio wave that can be reflected back by the ionosphere is called critical frequency.

Critical frequency, v_c = 9 (N_max)^{1 / 2}

Where, N_max = number density of electrons/metre³.

Skip Distance The minimum distance from the transmitter at which a sky wave of a frequency but not more than critical frequency, is sent back to the earth.

Skip distance (D_skip) = 2h (V_max / V_c)² – 1

where h is height of reflecting layer of atmosphere,

V_max is maximum frequency of electromagnetic waves and V_c is critical frequency.

Fading The variation in the strength of a signal at receiver due to interference of waves, is called fading.

(iii) Space Wave Propagation It is suitable for 30 MHz to 300 MHz. It is used in television communication and radar communication. It is also called line of sight communication.

• Range is limited due to curvature of earth. If h be the height of the transmitting antenna, then signal can be received upto a maximum distance

d = √2RH

• If height of transmitting and receiving antennas be h_T and h_R respectively. The effective range will

Microwave Propagation

• Microwaves are electromagnetic wave of frequency 1 to 300 GHz, greater than those of TV signals. The wavelength of microwaves is of the order of a few mm.
• Microwave communication is used in radar to locate the flying objects In space.
• These waves can be transmitted as beam signals in a particular direction, much better than radiowave,
• There is no diffraction of microwave around corners of an obstacle which happens to lie along its passage.

Satellite Communication

It IS carried out between a transmitter and a receiver through a satellite. A geostationary satellite is utilized for this purpose, whose time period is 24 hours.

A communication satellite is a space craft, provided with microwave receiver and transmitter. It is placed in an orbit around the earth. The India remote sensing satellites are

IRS-IA, IRS-IB and IRS-IC

The line-of-sight microwave communication through satellite is possible if the communication satellite is always at a fixed location with respect to the earth, e.g., the satellite which is acting as a repeater must be at rest with respect to the earth. It is so far a satellite known as geo-stationary satellite.

The basic requirements for geostationary satellites are as follows:

1. The time period of revolution of the satellite around the earth is equal to the time period of rotation of earth about its polar axis i.e., 24 h.
2. The sense of revolution of the satellite around the earth is the same as that of the earth about its polar axis i.e., from west to east.
3. The orbital plane of revolution of satellite is concentric and coplanar with the equatorial plane of earth.
4. The height of geostationary satellite above the equator of earth is nearly 36000 km and its orbital velocity is nearly 3.1 km/s.

The orbit in which the gee-satellite above revolves around the earth is known as geo-synchronous orbit. As its angular speed is synchronised with the angular speed of the earth. therefore, the geo-stationary satellite is also known as geo-synchronous satellite.

Merits of Satellites Communication

1. The satellite communication covers wide area for broadcasting a8 compared to other communication systems i.e. it has wide coverage range.
2. The satellite communication is also used effectively in mobile communication.
3. The satellite communication is found to be much economical as compared to other communication systems on earth. Infact. the cost involved in satellite communication is independent of the distance.
4. The satellite communication is most cost effective in remote and hilly areas, such as Ladakh, Himachal Pradesh etc.
5. The satellite communication permits transmission of data at rate.
6. The satellite communication is very accurate and economical search. rescue and navigation purposes.

Demerits of Satellite Communication

1. If a system on the satellite goes out of order due to environmental stresses, it is almost impossible to repair it.

2. In satellite communication, there is a time delay between transmission and reception, due to extremely large communication path length (greater than 2 x 36000 km). This delay causes a time gap during talking, which proves quite annoying.

Remote Sensing

It is a technique of observing or measuring the characteristics of the object at a distance. A polar satellite is utilized for this purpose.

Distance upto which a signal can be obtained from an antenna is given by

d = √2hR

where, h is height of antenna and R is radius of earth.

LED and Diode Laser in Communication

Light Emitting. Diode (LED) and diode laser are preferred sources for optical communication links to the following features.

1. Each produces light of suitable power required in optical communication. Diode laser provides light which is monochromatic and coherent. This light is obtained as a parallel beam. It is used in very long distance transmission.
2. LED provides almost monochromatics light. This suitable for small distance transmission. It is infact, a low cost device as compared to diode lasers.

Line Communication

• Transmission lines are used to interconnect points separated from each other. For example interconnection between a transmitter and a receiver or a transmitter and antenna or an antenna and a receiver are achieved through transmission lines.
• Line communication may be in the form of electrical signal or optical signal.

Optical Fibres

An optical fibre is a long thread consisting of a central core of glass or plastic of uniform refractive index. It is surrounded by a cladding of material of refractive index less than that of the core and a protective Jacket of insulating material.

There are three types of optical fibre configuration

1. Single mode step index fibre
2. Multi mode step index fibre
3. Multi mode graded index fibre.

Applications of Optical Fibres

1. A bundle of optical fibres is called light pipe. This pipe can transmit as image. Since the pipe is flexible, it can be twisted in any desired manner. Hence it is used medical and optical examination of even the inaccessible parts of human body, e.g., in endoscopy.
2. Optical fibres are used in transmission and reception of electrical signals by coverting them first into light signals.
3. Optical fibres are used in telephone and other transmitting cables. Each fibre can carry upto 2000 telephone messages without much loss of intensity.

Classification of substances on the basis of conduction of electricity.

Solid

We know that, each substance is composed of atoms. Substances are mainly classified into three categories namely solids, liquids and gases.

In each solid atoms are at a definite positions and the average distance between them is constant.

Depending upon the internal arrangement of atoms, solids are further divided into two groups.

1. Crystalline Solids

The solid in which the atoms are arranged in a regular order are called the crystalline solids. In other words, we can say that in a crystalline solid.there is periodicity and regularity of its component atoms in all the directions. For example sodium chloride (common salt), diamond, Sugar, silver etc are the crystalline solids.

Their atoms are arranged in a definite geometrical shape.

They have a definite melting point.

They are anisotropic, i.e., their physical properties such as thermal Conductivity refractive index etc, are different in different directions.

They are the real solids.

2. Amorphous Solids

The Solids in which the atoms do not have a definite arrangement are called the amorphous solids. They are also called the glassy solids. For example glass, rubber, plastic, power, etc are the amorphous solids.

Tbey do not have a definite arrangement of its atoms, i e., they do not have a characteristic geometrical shape.

They do not have a definite melting point.

They are isotropic. i.e., their physical properties such as conductivity of heat refractive index etc, are same in all the directions.

They are not the real solids.

Monocrystal and Polycrystalline

Monocrystal is a crystal in which the ordered arrangements of the atoms or molecules extends throughout the piece of solid, irrespective of its size.

Polycrystal is a crystalline solid in which each piece of the solid has a number of monocrystals with developed faces joined together.

The polycrystal ceramic made from PbO, ZnO and TiO are used in gas lighters and telephone receivers.

Liquid Crystals

Some organic crystalline solid. when heated acquire fluidity but retain their anisotropic properties. They axe called liquid crystals.

Some liquid crystals like cyanobiphenyl can change the plane of polarization of light and such Liquid Crystal Displays (LCD) are used in watches and micro calculators.

Crystal Lattice

A crystal is made up of a three- dimensional array of points such that each point is surrounded by the height bouring POints in an identical way. Such an array of points is known as bravais lattice or space lattice.

Unit cell is the smallest unit of the crystal lattice, repetition of which in three dimensions gives rise to crystal lattice.

The length of three sides of a unit cell are called Primitives or lattice constant represented by a, b, c. The angle between three crystallographic axis are called interfacial angles represented by α, β and γ. The primitives and interfacial angles constitute the lattice parameters of a unit cell.

[The cubic crystal may be of the form, simple cubic (sc) lattice, the body centred cubic (bee) lattice, the face centred Cubic (fcc) lattice.]

The coordination number is defined as the number of nearest neighbours around any lattice point (or atom) in the crystal lattice.

(a) For sc, coordination number is 6.

(b) For bee, coordination number is 8.

(c) For fcc, coordination number is 12.

(d) For sc, atomic radius is a / 2.

(e) For bcc, atomic is a √3 / 4.

(f) For fcc. atomic radius is a / 2√2.

Classification of solids on the basis of conductivity

(i) Conductor Conductors are those substances through which electricity can pass easily, e.g., all metals are conductors.

(ii) Insulator Insulators are those substances through which electricity cannot pass, e.g., wood. rubber, mica etc.

(iii) Semiconductor Semiconductors are those substances whose conductivity lies between conductors and insulators. e.g., germanium, silicon, carbon etc.

Energy Bands of Solids

1. 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 levels takes place. The collection of these closely spaced energy levels is called an energy band.

2. Valence Band

This energy band contains valence electrons. This band may be PartIally or completely filled with electrons but never be empty. The electrons in this band are not capable of gaining energy from external electric field to take part in conduction of current.

3. Conduction Band

This band contains conduction electrons. This band is either empty or Partially filled with electrons.

Electrons present in this band take part in the conduction of current.

4. Forbidden Band

This band is completely empty. The minimum energy required to shift an electron from valence band to conduction band is called band gap (E_g).

Thermionic Emission

Thermionic emission occurs when a metal is heated to a high temperature, the free electrons in the metal gain kinetic energy sufficient to escape through the surface of the metal.

Thermionic Diode

The thermionic diode is a two electrode (cathode and plate) device based on thermionic emission.

A diode allows unidirectional flow of electrons, i.e., only when the plate is positive with respect to cathode. Hence, it is also called a valve.

The triode value consists of three electrodes, e.g., cathode. plate and grid enclosed in an evacuated glass bulb.

Grid influences the space charge and controls the flow of plate current.

[When the grid is given a negative potential with respect to cathode. It repels the electrons escaping from the cathode and Increases the effect of space charge, at sufficientLy negative grid potential is known as cut-off grid bias.

If the grid is given a positive potential with respect to cathode, it attracts the electrons and decreases the effect of space charge. The increasing the plate current. In this case a current flow into the circuit, thus grid modifies the function of valve.]

Grid is always kept at small negative potential with respect to cathode.

Triode can be used as an amplifier, oscillator modulator and demodulator.

An oscillator is an electronic device which generates AC voltage froID DC power. It is basically a positive feedback amplifier with infinite voltage gain.

Types of Semiconductor

(i) Intrinsic Semiconductor A semiconductor in its pure state is called intrinsic semiconductor.

(ii) Extrinsic Semiconductor A semiconductor doped with suitable impurity to increase its impurity, is called extrinsic semiconductor.

On the basis of doped impurity extrinsic semiconductors are of two types

(i) n-type Semiconductor Extrinsic semiconductor doped with pentavalent impurity like As, Sb, Bi, etc in which negatively charged electrons works as charge carrier, is called n-type semiconductor.

Every pentavalent impurity atom donate one electron in the crystal, therefore it is called a doner atom

(ii) p -type Semiconductor Extrinsic semiconductor doped with trivalent impurity like Al, B, etc, in which positively charged holes works as charge carriers, is called p-type semiconductor.

Every trivalent impurity atom have a tendency to accept one electron, therefore it is called an acceptor atom.

In a doped semiconductor n_e n_h = n²_i where n_e and n_h are the number density of electrons and holes and n_i is number density of intrinsic carriers, i.e., electrons or holes.

In n-type semiconductor, n_e > > n_h

In p -type semiconductor, n_h > > n_e

Electrical conductivity of extrinsic semiconductor is given by

σ = 1 / ρ = e (n_e μ_e + n_h μ_h)

where ρ is resistivity, μ_e and μ_h are mobility of electrons and holes respectively.

Note Energy gap for Ge is 0.72 eV and for Si it is 1.1 eV.

p-n Junction

An arrangement consisting a p -type semiconductor brought into a close contact with n-type semiconductor, is called a p -n junction.

The current in a p-n junction is given by

k_B = I_o (e^{eV/k BT} – 1 )

where I_o is reverse saturation current, V is potential difference across the diode, and k_B is the Boltzmann constant.

Terms Related to p-n Junction

(i) Depletion Layer At p-n. junction a region is created, where there is no charge carriers. This region is called depletion layer. The width of this region is of the order of 10⁶ m.

(ii) Potential Barrier The potential difference across the depletion layer is called potential barrier.

Barrier potential for Ge is 0.3 V and for Si is 0.7 V.

(iii) Forward Biasing In this biasing, the p -side is connected to positive terminal and n-side to negative terminal of a battery.

In this biasing, forward current flows due to majority charge carriers.

The width of depletion layer decreases.

(iv) Reverse Biasing In this biasing, the p-side is connected to negative terminal and n-side to positive terminal of a battery.

In this biasing, reverse current flows due to minority charge carriers.

The width of depletion layer increases.

A p-n junction diode can be utilized as a rectifier.

Zener diode, photo-diode, light-emitting diode, etc are specially designed p-n. junction diodes.

p-n Junction Diode

The current through p-n junction flow only from p toward n and not from n toward p.

The maximum voltage that a junction diode can bear without break is called zener voltage and the junction diodes possessing this voltage is known as zener diode.

Resistance of diode R = V / I

Rectifier

A device which convert alternating current or voltage into direct current or voltage IS known as rectifier. The process of converting AC into DC IS caned rectification.

Half-Wave Rectifier

A half-wave rectifier converts the half cycle of applied AC signal into DC signal. Ordinary transformer may be used here.

Full-Wave Rectifier

A full-wave rectifier converts the whole cycle of applied AC signal into DC signal. Centre top,transformer is used here.

[Half-wave rectifier converts only one-half of AC Into DC while full wave rectifier rectifies both halves of AC input.]

Transistor

A transistor is an arrangement obtained by growing a thin layer of one type of semiconductor between two thick layers of other similar type semiconductor.

Types of Transistors

• The left side semiconductor is called emitter, the right side semiconductor is called collector and the thin middle layer is called base.
• Emitter is highly doped and base is feebly doped.
• A transistor can be utilized as an amplifier and oscillator but not a rectifier
• Maximum amplification is obtained in common-emitter configuration.

Transistor as an Amplifier

An amplifier is a device which is used for increasing the amplitude of variation of alternating voltage 01′ current or power.

The amplifier thus produces an enlarged version of the input signal.

The general concept of amplification is represented in figure. There are two input terminals for the signal to be amplified and two output terminals for connecting the load; and’ a means of supplying power to the amplifier.

1. In Common Base Amplifier,

AC current gain (α_AC) = ΔI_c / ΔI_e

where ΔI_c is change in collector current and ΔI_e change in emitter current.

AC voltage gain (A_V) = Output voltage / Input voltage

= α_AC * Resistance gain = α_AC * R_o / R_i

where R_o is output resistance of the circuit and R_i is input resistance of the circuit.

AC power gain = Change in output power / Change in input power

= AC voltage gain * AC current gain

= α²_AC * resistance gain

The input and output signals are in the same phase.

There is no amplification in current of a given signal.

There is an amplification in voltage and power of the given signal.

2. In Common Emitter Amplifier

AC current gain (&beta_AC) = ΔI_c / ΔI_e

where ΔI_c is change in collector current and ΔI_e change in base current.

AC voltage gain (A_V) = &beta_AC * resistance gain

AC power gain =&beta²_AC * Resistance gain

Relation between the current gain of common base and common emitter amplifier.

β = α / 1 – α = I_c / I_e

The input and output signals are out of phase by π or 180°

There is amplification in current, voltage and power of the given signal.

Light Emitting Diodes (LED)

It is forward biased p-ri junction diode which emits light when recombination of electrons and holes takes place at the junction.

If the semiconducting material of p-ri junction is transparent to light, the light is emitting and the junction becomes a light source, i.e., Light Emitting Diode (LED).

The colour of the light depends upon the types of material used in making the semiconductor diode.

(i) Gallium – Arsenide (Ga-As) – Infrared radiation

(ii) Gallium – phosphide (GaP) – Red or green light

(iii) Gallium – Arsenide – phosphide (GaAsP) – Red or yellow light

Logic Gate

A digital circuit which allows a signal to pass through it, only when few logical relations are satisfied, is called a logic gate.

Truth Table

A table which shows all possible input and output combinations is called a truth table.

Basic Logic Gates

(i) OR Gate It is a two input and one output logic gate.

(ii) AND Gate It is a two input and one output logic gate

(iii) NOT Gate It is a one input and one output logic gate.

Combination of Gates

(i) NAND Gate When output of AND gate is applied as input to a NOT gate, then it is called a NAND gate.

Boolean expression Y = A * B (Y equals negated of A AND B)

(ii) NOR Gate When output of OR gate is applied as input to a NOT gate, then it is called a NOR gate.

Boolean expression Y = A + B ( Y equals negated of A OR B)

• The Boolean expression obey commutative law associative law as well as distributive law.

1. A + B = B+ A
2. A· B = B· A
3. A + (B + C) = (A + B) + C

• Demorgan’s theorems

1. A + B = A * B
2. A * B = A + B

The entire positive charge and nearly the entire mass of atom is concentrated in a very small space called the nucleus of an atom.

The nucleus consists of protons and neutrons. They are called nucleons.

Terms Related to Nucleus

(i) Atomic Number The number of protons in the nucleus of an atom of the element is called atomic number (Z) of the element.

(ii) Mass Number The total number of protons and neutrons present inside the nucleus of an atom of the element is called mass number (A) of the element.

(iii) Nuclear Size The radius of the nucleus R ∝ A^1/3

⇒ R = R_o A^1/3

where, R_o = 1.1 * 10^-15 m is an empirical constant.

(iv) Nuclear Density Nuclear density is independent of mass number and therefore same for all nuclei.

ρ = mass of nucleus / volume of nucleus ⇒ ρ = 3m / 4π R³_o

where, m = average mass of a nucleon.

(v) Atomic Mass Unit It is defined as 1 / 12th the mass of carbon nucleus.

It is abbreviated as arnu and often denoted by u. Thus

1 amu = 1.992678 * 10^-26 / 12 kg

= 1.6 * 10^-27 kg = 931 Me V

Isotopes

The atoms of an element having same atomic number but different mass numbers. are called isotopes.

e.g., ₁H¹, ₁H², ₁H³ are isotopes of hydrogen.

Isobars

The atoms of different elements having same mass numbers but different atomic numbers, are called isobars.

e.g., ₁H³, ₂He³ and ₁₀Na²², ₁₀Ne²² are isobars.

Isotones

The atoms of different elements having different atomic numbers and different mass numbers but having same number of neutrons, are called isotones.

e.g., ₁H³, ₂He⁴ and ₆C¹⁴, ₈O¹⁶ are isobars.

Isomers

Atoms having the same mass number and the same atomic number but different radioactive properties are called isomers,

Nuclear Force

The force acting inside the nucleus or acting between nucleons is called nuclear force.

Nuclear forces are the strongest forces in nature.

• It is a very short range attractive force.
• It is non-central. non-conservative force.
• It is neither gravitational nor electrostatic force.
• It is independent of charge.
• It is 100 times that of electrostatic force and 10³⁸ times that of gravitational force.

According to the Yukawa, the nuclear force acts between the nucleon due to continuous exchange of meson particles.

Mass Defect

The difference between the sum of masses of all nucleons (M) mass of the nucleus (m) is called mass defect.

Mass Defect (Δm) = M – m = [Zm_p + (A – Z)m_n – m_n]

Nuclear Binding Energy

The minimum energy required to separate the nucleons up to an infinite distance from the nucleus, is called nuclear binding energy.

Nuclear binding energy per nucleon = Nuclear binding energy / Total number of nucleons

Binding energy, E_b = [Zm_p + (A – Z) m_n – m_N]c²

Packing Fraction (P)

p = (Exact nuclear mass) – (Mass number) / Mass number

= M – A / M

The larger the value of packing friction. greater is the stability of the nucleus.

[The nuclei containing even number of protons and even number of neutrons are most stable.

The nuclei containing odd number of protons and odd number of neutrons are most instable.]

Radioactivity

The phenomena of disintegration of heavy elements into comparatively lighter elements by the emission of radiations is called radioactivity. This phenomena was discovered by Henry Becquerel in 1896.

Radiations Emitted by a Radioactive Element

Three types of radiations emitted by radioactive elements

(i) α-rays

(ii) β-rays

(iii) γ – rays

α-rays consists of α-particles, which are doubly ionised helium ion.

β-rays are consist of fast moving electrons.

γ – rays are electromagnetic rays.

[When an α – particle is emitted by a nucleus its atomic number decreases by 2 and mass number decreases by 4.

When a β -particle is emitted by a nucleus its atomic number is Increases by one and mass number remains unchanged.

When a γ – particle is emitted by a nucleus its atomic number and mass number remain unchanged

Radioactive Decay law

The rate of disintegration of radioactive atoms at any instant is directly proportional to the number of radioactive atoms present in the sample at that instant.

Rate of disintegration ( – dN / dt) ∝ N

– dN / dt = λ N

where λ is the decay constant.

The number of atoms present undecayed in the sample at any instant N = N_o e^-λt

where, N_o is number of atoms at time t = 0 and N is number of atoms at time t.

Half-life of a Radioactive Element

The time is which the half number of atoms present initially in any sample decays, is called half-life (T) of that radioactive element.

Relation between half-life and disintegration constant is given by

T = log²_e / λ = 0.6931 / λ

Average Life or Mean Life(τ)

Average life or mean life (τ) of a radioactive element is the ratio of total life time of all the atoms and total number of atoms present initially in the sample.

Relation between average life and decay constant τ = 1 / λ

Relation between half-life and average life τ = 1.44 T

The number of atoms left undecayed after n half-lifes is given by

N = N_o (1 / 2)ⁿ = N_o (1 / 2) ^t/T

where, n = t / T, here t = total time.

Activity of a Radioactive Element

The activity of a radioactive element is equal to its rate of disintegration.

Activity R = ( – dN / dt)

Activity of the sample after time t,

R = R_o e ^-λt

Its SI unit is Becquerel (Bq).

Its other units are Curie and Rutherford.

1 Curie = 3.7 * 10¹⁰ decay/s

1 Rutherford = 10⁶ decay/s

Nuclear Fission

The process of the splitting of a heavy nucleus into two or more lighter nuclei is called nuclear fission.

When a slow moving neutron strikes with a uranium nucleus (₉₂U²³⁵), it splits into ₅₆Ba¹⁴¹ and ₃₆Kr⁹² along with three neutrons and a lot of energy.

Nuclear Chain Reaction

If the particle starting the nuclear fission reaction is produced as a product and further take part in the nuclear fission reaction, then a chain of fission reaction started, which is called nuclear chain reaction.

Nuclear chain reaction are of two types

(i) Controlled chain reaction

(ii) Uncontrolled chain reaction

Nuclear Reactor

The main parts of a nuclear reactor are following

(i) Fuel Fissionable materials like ₉₂U²³⁵, ₉₂U²³⁸, ₉₄U²³⁹ are used as fuel.

(ii) Moderator Heavy water, graphite and beryllium oxide are used to slower down fast moving neutrons.

(iii) Coolant The cold water, liquid oxygen, etc. are used to remove heat generated in the fission process.

(iv) Control rods Cadmium or boron rods are good absorber of neutrons and therefore used to control the fission reaction.

Atom bomb working is based on uncontrolled chain reaction.

Nuclear Fusion

The process of combining of two lighter nuclei to form one heavy nucleus, is called nuclear fusion.

Three deuteron nuclei (₁H²) fuse, 21.6 MeV is energy released and nucleus of helium (₂He⁴) is formed.

In this process, a large amount of energy is released.

Nuclear fusion takes place at very high temperature approximately about 10⁷ K and at very high pressure 10⁶ atmosphere.

Hydrogen bomb is based on nuclear fusion.

The source of Sun’s energy is the nuclear fusion taking place at sun.

Thermonuclear Energy

The energy released during nuclear fusion is know as thermonuclear energy. Protons are needed for fusion while neutrons are needed for fission process.

All elements are consists of very small invisible particles, called atoms. Atoms of same element are exactly same and atoms of different element are different.

Thomson’s Atomic Model

Every atom is uniformly positive charged sphere of radius of the order of 10^-10 m, in which entire mass is uniformly distributed and negative charged electrons are embedded randomly. The atom as a whole is neutral.

Limitations of Thomson’s Atomic Model

1. It could not explain the origin of spectral series of hydrogen and other atoms.

2. It could not explain large angle scattering of α – particles.

Rutherford’s Atomic Model

On the basis of this experiment, Rutherford made following observations

(i) The entire positive charge and almost entire mass of the atom is concentrated at its centre in a very tiny region of the order of 10^-15 m, called nucleus.

(ii) The negatively charged electrons revolve around the nucleus in different orbits.

(iii) The total positive charge 011 nucleus is equal to the total negative charge on electron. Therefore atom as a overall is neutral.

(iv) The centripetal force required by electron for revolution is provided by the electrostatic force of attraction between the electrons and the nucleus.

Distance of Closest Approach

r_o = 1 / 4π ε_o . 2Ze² / E_k

where, E_k = kinetic energy of the cc-particle.

Impact Parameter

The perpendicular distance of the velocity vector of a-particle from the central line of the nucleus, when the particle is far away from the nucleus is called impact parameter.

Impact parameter

where, Z = atomic number of the nucleus, E_k = kinetic energy of the c-particle and θ = angle of scattering.

Rutherford’s Scattering Formula

where, N(θ) =number of c-particles, N_i = total number of α-particles reach the screen. n = number of atoms per unit volume in the foil, Z = atoms number, E = kinetic energy of the alpha particles and t = foil thickness

Limitations of Rutherford Atomic Model

(i) About the Stability of Atom According to Maxwell’s electromagnetic wave theory electron should emit energy in the form of electromagnetic wave during its orbital motion. Therefore. radius of orbit of electron will decrease gradually and ultimately it will fall in the nucleus.

(ii) About the Line Spectrum Rutherford atomic model cannot explain atomic line spectrum.

Bohr’s Atomic Model

Electron can revolve in certain non-radiating orbits called stationary or bits for which the angular momentum of electron is an integer multiple of (h / 2π)

mvr = nh / 2π

where n = I, 2. 3,… called principle quantum number.

The radiation of energy occurs only when any electron jumps from one permitted orbit to another permitted orbit.

Energy of emitted photon

hv = E₂ – E₁

where E₁ and E₂are energies of electron in orbits.

Radius of orbit of electron is given by

r = n²h² / 4π² mK Ze² ⇒ r ∝ n² / Z

where, n = principle quantum number, h = Planck’s constant, m = mass of an electron, K = 1 / 4 π ε, Z = atomic number and e = electronic charge.

Velocity of electron in any orbit is given by

v = 2πKZe² / nh ⇒ v ∝ Z / n

Frequency of electron in any orbit is given by

v = KZe² / nhr = 4π²Z²e⁴mK² / n³ h³

⇒ v prop; Z³ / n³

Kinetic energy of electron in any orbit is given by

E_k = 2π²me⁴Z²K² / n² h² = 13.6 Z² / n² eV

Potential energy of electron in any orbit is given by

E_p = – 4π²me⁴Z²K² / n² h² = 27.2 Z² / n² eV

⇒ E_p = ∝ Z² / n²

Total energy of electron in any orbit is given by

E = – 2π²me⁴Z²K² / n² h² = – 13.6 Z² / n² eV

⇒ E_p = ∝ Z² / n²

Wavelength of radiation emitted in the radiation from orbit n₂ to n₁ is given by

In quantum mechanics, the energies of a system are discrete or quantized. The energy of a particle of mass m is confined to a box of length L can have discrete values of energy given by the relation

E_n = n² h² / 8mL² ; n < 1, 2, 3,…

Hydrogen Spectrum Series

Each element emits a spectrum of radiation, which is characteristic of the element itself. The spectrum consists of a set of isolated parallel lines and is called the line spectrum.

Hydrogen spectrum contains five series

(i) Lyman Series When electron jumps from n = 2, 3,4, …orbit to n = 1 orbit, then a line of Lyman series is obtained.

This series lies in ultra violet region.

(ii) Balmer Series When electron jumps from n = 3, 4, 5,… orbit to n = 2 orbit, then a line of Balmer series is obtained.

This series lies in visual region.

(iii) Paschen Series When electron jumps from n = 4, 5, 6,… orbit to n = 3 orbit, then a line of Paschen series is obtained.

This series lies in infrared region

(iv) Brackett Series When electron jumps from n = 5,6, 7…. orbit to n = 4 orbit, then a line of Brackett series is obtained.

This series lies in infrared region.

(v) Pfund Series When electron jumps from n = 6,7,8, … orbit to n = 5 orbit, then a line of Pfund series is obtained.

This series lies in infrared region.

Wave Model

It is based on wave mechanics. Quantum numbers are the numbers required to completely specify the state of the electrons.

In the presence of strong magnetic field, the four quantum number are

(i) Principle quantum number (n) can have value 1,2, … ∞

(ii) Orbital angular momentum quantum number l can have value 0,1, 2, … ,(n – 1).

(iii) Magnetic quantum number (m_e) which can have values – I to I.

(iv) Magnetic spin angular momentum quantum number (m_s) which can have only two value + 1 / 2.

Cathode rays are the stream of fast moving electrons. These rays are produced in a discharge tube at a pressure below 0.01 rom of mercury.

Properties of Cathode Rays

(i) Cathode rays are not electromagnetic rays.

(ii) Cathode rays are deflected by electric field and magnetic field.

(iii) Cathode rays produce heat in metals when they fallon them.

(iv) Cathode rays can pass through thin aluminium or gold foils without puncturing them.

(v) Cathode rays can produce physical and chemical change.

(vi) Cathode ray travel in straight line with high velocity momentum and energy and cast shadow of objects placed in their path.

(vii) On striking the target of high atomic weight and high melting point, they produce X-rays.

(viii) Cathode rays produce fluorescence and phosphorescence in certain substance and hence affect photographic plate.

(ix) When any charge particle move in a field where magnetic and electric fields are present, without any deviation, then

Magnetic force = Electrostatic force

Bev = Ee or v = E / B

(x) Specific charge of cathode rays means the ratio of charge and mass.

(xi) Specific charge of electron was determined by J J Thomson using perpendicular magnetic and electric field applied on a beam of electrons, at the same place.

(xii) Specific charge of electron e / m = E² / 2VB²

where, E = electric field, B = magnetic field and V = potential difference applied across ends of tube.

(xii) The value of specific charge of an electron is 1.7589 * 10¹¹ C / kg.

(xiv) Millikan measured the charge of an electron through his popular oil drop experiment.

(xv) The charge of the electron as determined by Millikan was found to be 1.602 * 10^-19 C.

Positive Rays

Positive rays were discovered by Goldstein. Positive rays are moving positive ions of gas filled in the discharge tube. The mass of these particles is nearly equal to the mass of the atoms of gas.

(i) These consist of fast moving positively charged particles.

(ii) These rays are deflected in magnetic and electric fields.

(iii) These rays travel in straight line.

(iv) Speed of positive rays is less than that of cathode rays.

(v) These rays can produce fluorescence and phosphorescence.

Electron Emission

It is the phenomenon of emission of electron from the surface of a metal. The electron emission can be obtained from the following process

(i) Thermionic

(ii) Photoelectric emission

(iii) Field emission

(iv) Secondary emission

Photon

Photons are the packets of energy emitted by a source of radiation. The energy of each photon is,

E = hv

Where h is Planck’s constant and v is frequency of radiation.

The rest mass of a photon is zero.

The momentum of a photon p = hv / c = h / λ

Dynamic or kinetic mass 0f photon m = hv / c² = h / cλ

where c is speed of light in vacuum and λ is wavelength of radiation. Photons are electrically neutral.

A body can radiate or absorb energy in whose number multiples of a quantum hv, 2hv, 3hv …. nhv, where n is positive integer.

Photoelectric Effect

The phenomena of emission of electrons from a metal surface, when radiations of suitable frequency is incident on it, is called photoelectric effect.

Terms Related to Photoelectric Effect

(i) Work Function(φ) The minimum amount of energy required to eject one electron from a metal surface, is called its work function.

(ii) Threshold Frequency (v_o) The minimum frequency of light which can eject photo electron from a metal surface is called threshold frequency of that metal.

(iii) Threshold Wavelength (λ_max) The maximum wavelength rJ light which can eject photo electron from a metal surface is called threshold wavelength of that metal.

Relation between work function, threshold frequency and threshold wavelength

φ = hv_o = hc / λ_max

Laws of Photoelectric Effect

1. For a given metal and frequency of incident light, the photo electric current (the rate of emission of photoelectrons) is directly proportional to the intensity of incident light.

2. For a given metal. there is a certain minimum frequency, called threshold frequency, below which there is no emission of photo electrons takes place.

3. Above threshold frequency the maximum kinetic energy of photo electrons depends upon the frequency of incident light.

4. The photoelectric emission is an instantaneous process.

Einstein’s Photoelectric Equation

The maximum kinetic energy of photoelectrons

(E_k)_max = hv – φ = h(v – v_o)

where v is frequency of incident light and v_o is threshold frequency.

Stopping Potential

The minimum negative potential given to anode plate at which photoelectric current becomes zero is called stopping potential (V_o).

Maximum kinetic energy of photo electrons

(E_k)_max = 1 / 2 mv²_max = eV_o

Compton Effect

When a monochromatic beam of X – falls on a target containing free electrons. it is scattered. As a result, the electrons recoil and scattered radiation has wavelength longer than incident one. This effect is called Compton effect.

(i) λ’ – λ = λ = Compton shift Δλ = h / m_oc (1 – cos φ) where m_o is rest mass of an electron and c is the speed of light h / m_oc Compton shift Δλ is maximum, when φ = 180°

(ii) Kinetic energy of recoil electron

E_k = hc / λ – hc / λ’

(iii) Direction of recoil electron

tan θ = λ sin φ / λ’ – λ cos φ

(iv) Compton wavelength of electron

= h / m_oc = 0.024 A

(v) Maximum Compton shift

(Δλ)_max = 2h / m_oc 0.0048 A

Matter Waves on de-Broglie Waves

A wave is associated with every moving particle, called matter or de-Broglie wave.

de-Broglie Wavelength

If a particle of mass m is moving with velocity v, then wavelength of de-Broglie wave associated with it is given by

λ = h / p = h / mv

de-Broglie wavelength of an electron is given by

λ = h / mv = h / √2me V = 12.27 / √V A.

where, m = mass of electron, e = electronic charge and V = potential difference with which electron is accelerated.

Davisson and Germer proves the existence of de-Broglie waves associated with an electron in motion.

Davisson-Germer Experiment

The wave nature of the material particles as predicted by de-Broglie was confirmed by Davisson and Germer (1927) in united states and by GP Thomson (1928) in scotland.

This experiment verified the wave nature of electron using Ni crystal.

Davisson and Germer found that the intensity of scattered beam of electrons was not the same but different at different angles of scattering. It is maximum for diffracting angle 50° at 54 V potential difference.

X-rays

When cathode rays strike on a heavy metal of high melting point. then a very small fraction of its energy converts in to a new type of waves, called X-rays.

Properties of X-rays

X-rays were discovered by Roentgen.

(i) X-rays are electromagnetic waves of wavelengths ranging from 0.1 A to 100 A and frequencies ranging from 10¹⁶ Hz to 10¹⁸ Hz.

(ii) Soft X-rays have greater wavelength and lower frequency.

(iii) Hard X-rays have lower wavelength and higher frequency.

(iv) X-rays are produced by coolidge tube.

(v) Molybdenum and tungsten provide suitable targets. These elements have large atomic number and high melting point for the purpose.

(vi) The intensity of X – rays depends on the heating voltage or filament current.

(vii) The kinetic energy of X-ray photons depends upon the voltage applied across the ends of coolidge tube.

(viii) Energy of X-ray photon is given by E = hv = hc / λ

(ix) If total energy of fast moving electron transfer to X-ray photon, then its energy, eV = hv = hc / λ

(x) Wavelength of emitted X-rays is given by λ = hc / eV

where, h = Planck’s constant, c = speed of light, e = electronic charge and V = potential difference applied across the ends of the tube.

(xi) Absorption of X-rays

I = I_oe^{– μx}, where I_o = initial intensity of X-rays, I = final intensity of emergent X-rays, x = thickness of material and μ = absorption coefficient.

Diffraction of X-rays

X-rays can be diffracted by crystals following Bragg’s law. According to which

2d sin θ = n λ

where, n = 1, 2, 3, …, and d = spacing of crystal planes, θ = angle of diffraction.

X-rays Spectrum

The energy spectrum of X-. rays is a line spectrum, containing following series :

(i) K – series When electrons of any higher orbit (n = 2,3,4, … ) jump to first orbit (n = 1) then K-series of X-rays are produced.

(ii) L – series When electrons of higher orbit (n = 3, 4, 5, … ) jump to second orbit (n = 2), then L-series of X-rays are produced.

(iii) M – series When electrons of higher orbit (n = 4,5,6, … )jump to third orbit (n = 3), then M-series of X-rays are produced.

First lines of these series are called K_α, L_α, M_α. Second lines of these series are called K_β, L_β, M_β

Moseley’s Law

The frequency of X-ray is given by

V = a (Z – b)²

where a and b are constants and Z is atomic number of element.

Frequency of X-rays

v ∝ Z²

Newton’s Corpuscular Theory

Light consists of very small invisible elastic particles which travel in vacuum with a speed of 3 x 10⁸ m/s.

The theory could explain reflection and refraction.

The size of corpuscular of different colours of light are different.

It could not explain interference, diffraction, polarisation. photoelectric effect and Compton effect. The theory failed as it could not explain why light travels faster in a rarer medium than in a denser medium.

Wavefront

A wavefront is defined as the continuous locus of all the particles of a medium, which are vibrating in the same phase.

These are three types

(i) Spherical wavefront

(ii) Cylindrical wavefront

(iii) Plane wavefront

Huygen’s Wave Theory

Light travel in a medium in the form of wavefront.

A wavefront is the locus of all the particles vibrating in same phase.

All particles on a wavefront behaves as a secondary source of light, which emits secondary wavelets.

The envelope of secondary wavelets represents the new position of a wavefront.

When source of light is a point source,the wavefront is spherical.

Amplitude (A) is inversely proportional to distance (x) i.g., A ∝ 1 / x .

∴ Intensity (I) ∝ (Amplitude)²

When SOurce of light is linear, the wavefront is cylindrical.

Amplitude (A) ∝ 1 / √x

∴ Intensity ∝ (Amplitude)² ∝ 1 / x

Huygen’s Principle

(i) Every point on given wavefront (called primary wavefront) acts as a fresh source of new disturbance called secondary wavelets.

(ii) The secondary wavelets travels in all the directions with the speed of light in the medium.

(iii) A surface touching these secondary wavelets tangentially in the forward direction at any instant gives the new (secondary) wave front of that instant.

Maxwell’s Electromagnetic Wave Theory

(i) Light waves are electromagnetic waves which do not require a material medium for their propagation.

(ii) Due to transverse nature, light wave undergo polarisation.

(iii) The velocity of electromagnetic wave in vacuum is c = 1 / √μ_o ε_o

(iv) The velocity of electromagnetic waves in medium is less than that of light, v < c

v = 1 / √μ_o ε_o ε_r μ_r = c / √μ_o ε_r

(v) The velocity of electromagnetic waves in a medium depend upon the electric and magnetic properties of the medium.

where, μ_o = absolute magnetic permeability and

ε_o = absolute electrical permittivity of free space.

(vi) It failed to explain the phenomenon of photoelectric effect, Compton effect and Raman effect.

Max Planck’s Quantum Theory

(i) Light emits from a source in the form of packets of energy called quanta or photon.

(ii) The energy of a photon is E == hv, where h is Planck’s constant and v is the frequency of light.

(iii) Quantum theory could explain photoelectric effect, Compton effect and Raman effect.

(iii) Quantum theory failed to explain interference, diffraction and polarisation of light.

de – Broglie’s Dual Theory

Light waves have dual nature, wave nature according to Maxwell’s electromagnetic wave theory and particle nature according to Max-Planck’s quantum theory.

Two natures of light are like the two faces of a coin. In anyone phenomena only its one nature appears.

Energy of photon = hv = hc / λ

where, h = Planck’s constant 6.6 * 10<sup-34 J / s

de-Broglie wave equation is λ = h / p = h / mv

where h denotes Planck’s constant.

Superposition of Waves

When two similar waves propagate in a medium simultaneously, then at any point the resultant displacement is equal to the vector sum of displacement produced by individual waves.

y = y₁ + y₂

Interference of Light

When two light waves of similar frequency having a zero or constant phase difference propagate in a medium simultaneously in the same direction, then due to their superposition maximum intensity is obtained at few points and minimum intensity at other few points.

This phenomena of redistribution of energy due to superposition of waves is called interference of light waves.

The interference taking place at points of maximum intensity is called constructive interference.

The interference taking place at points of minimum intensity is destructive interference.

Fringe Width

The distance between the centres of two consecutive bright or dark fringes is called the fringe width.

The angular fringe width is given by θ = λ / d.

where λ is the wavelength of light d is the distance between two coherent sources.

Conditions for Constructive and Destructive Interference

For Constructive Interference

Phase difference, φ = 2nπ

Path difference, Δx = nλ

where, n = 0, 1, 2, 3,…

For Destructive Interference

Phase difference, φ = (2n – 1)π

Path difference, Δx = (2n – 1)π / 2

where, n = 1, 2, 3, …

If two waves of exactly same frequency and of amplitude a and b interfere, then amplitude of resultant wave is given by

R = √a² + b² + 2ab cos φ

where φ is the phase difference between two waves.

R_max = (a + b)

R_min = (a – b)

Intensity of wave

∴ I = a² + b² + 2ab cos φ

= I₁ + I₂ + 2 √I₁ I₂ cos φ

where I₁ and I₂ are intensities of two waves.

∴ I₁ / I₂ = a² / b² = ω₁ / ω₂

Where ω₁ and ω₂ are width of slits.

Energy remains conserved during interference.

Interference fringe width

β = Dλ / d

where, D = distance of screen from slits, λ = wavelength of light and d = distance between two slits.

Distance of nth bright fringe from central fringe x_n = nDλ / d

Distance of nth dark fringe from central fringe x’_n = (2n – 1) Dλ / 2d

Coherent Sources of Light

The sources of light emitting light of same wavelength, same frequency having a zero or constant phase difference are called coherent sources of light.

When a transparent sheet of refractive index μ and of thickness t is introduced in one of the path of interfering waves, then fringe pattern shifts in that direction by a distance Y

Y = D / d (μ – 1) t = β / λ (μ – 1) t

where, β = fringe width.

Fresnel’s Biprism

It is a combination of two prisms of very small refracting angles placed base to base. It is used to obtain two coherent sources from a single light source.

Lyod’s Mirror

The shape of interference fringes are usually hyperbolic.

When screen is held at 900 to the line joining foci of the hyperbola, the fringes are circular.

When distance of screen (D) is very large compare to the distance between the slits (d), the Cringes are straight.

Diffraction

The bending of light waves around the corners of an obstacle or aperture is called diffraction of light.

The phenomenon of diffraction is divided mainly in the following two classes

(a) Fresnel class
(b) Fraunhofer class

Fraunhofer Diffraction at a Single Slit

Linear Width 0f central maximum 2Dλ / a = 2fλ / a

Angular width of central maximum = 2λ / a

where, λ = wavelength of light, a = width of single slit, D = distance of screen from the slit and f = focal length of convex lens.

For Secondary Minima

(a) Path difference = nλ

(b) Linear distance = nDλ / a = nfλ / a

(c) Angular spread = nλ / a

where, n = 1, 2, 3,.,.

For Secondary Maxima

(a) Path difference = (2n + 1 ) &lamda; / 2

(b) LInear distance = (2n + 1 ) D&lamda; / 2a = (2n + 1 ) f&lamda; / 2a

(c) Angular spread = (2n + 1 ) &lamda; / 2

Important Points

• A soap bubble or oil film on water appears coloured in white light due to interference of light reflected from upper and lower surfaces of soap bubble or oil film.
• In interference fringe pattern all bright and dark fringes are of same width,
• In diffraction fringe pattern central bright fringe is brightest and widest. and I remaining secondary maximas are of gradually decreasing intensities.
• The difference between interference and diffraction is that the interference is the superposition between the wavelets coming from two coherent sources while the diffraction is the superposition between the wavelets coming from the single wavefront

Polarisation

The phenomena of restructuring of electric vectors of light into a single direction is called polarisation.

Ordinary light has electric vectors in all possible directions in a plane perpendicular to the direction of propagation of light.

When ordinary light is passed through a tourmaline, calcite or quartz crystal the transmitted light have electric vectors in a particular direction parallel to the axis of crystal. This light is plane polarised light.

[A plane containing the vibrations of polarised light is called plane of vibration.

A plane perpendicular to the plane of vibration is called plane of polarisation.]

Polarisation can take place only in transverse waves.

Nicol Prism

A nicol prism is an optical device which is used for producing   plane polarised light and analysing light the same.

The nicol prism consists of two calcite crystal cut at 68° with its principal axis joined by a glue called Canada balsam.

Law of Malus

When a beam of completely plane polarised light is incident on an analyser, the intensity of transmitted light from analyser is directly proportional to the square of the cosine of the angle between plane of transmission of analyser and polariser, i.e.,

I ∝ cos² θ

When ordinary light is incident on a polariser the intensity of transmitted light is half of the intensity of incident light.

When a polariser and analyser are perpendicular to each other, then intensity of transmitted light from analyser becomes O.

Brewster’s Law

When unpolarised light is incident at an angle of polarisation (i_p) on the interface separating air from a medium of refractive index μ, then reflected light becomes fully polarised, provided

μ = tan i_p

If angle of polarisation is i_p and angle of refraction is μ then

i_p + r = 90°

Refractive index μ = tan i_p = 1 / sin C

where, C = critical angle.

Double Refraction

When unpolarised light is incident on a calcite or quartz crystal it splits up into two refracted rays. one of which follows laws of refraction. called ordinary ray (O-ray) and other do not follow laws of refraction. called extraordinary ray (E-ray). This phenomena is called double refraction.

Dichroism

Few double refracting crystals have a property of absorbing one of the two refracted rays and allowing the other to emerge out. This property of crystal is called dichroism.

Polaroid

It is a polarising film mounted between two glass plates. It is used to produce polarised light.

A polaroid is used to avoid glare of light in spectacles.

Uses of Polaroid

(i) Polaroids are used in sun glasses. They protect the eyes from glare.
(ii) The polaroids are used in window panes of a train and especially of an aeroplane. They help to control the light entering through the window.
(iii) The pictures taken by a stereoscopic camera. When seen with the help of polarized spectacles, create three dimensional effect.
(iv) The windshield of an automobile is made of polaroid. Such a mind shield protects the eyes of the driver of the automobile from the dazzling light of the approaching vehicles.

Light is a form of energy eyes. which produces the Sources of light are of three types-thermal sources and luminescent sources.

Photometry is a branch measurement of light energy.

Characteristics of Light

Light waves are electromagnetic waves, whose nature is transverse. The speed of light in vacuum is 3 x 10⁸ mls but it is different in different media.

The speed and wavelength of light change when it travels from one medium to another but its frequency remains unchanged.

Important Terms

(i) Luminous Objects The objects which emits its own light, are called luminous objects, e.g., sun, other stars, an oil lamp etc.

(ii) Non-Luminous Objects The objects which do not emit its own light but become visible due to the reflection of light falling on them, are called non-luminous objects, e.g., moon, table, chair. trees etc.

(iii) Ray of Light A straight line drawn in the direction of propagation of light is called a ray of light.

(iv) Beam of Light A bundle of the adjacent light rays is called a beam of light.

(v) Image If light ray coming from an object meets or appear to meet at a point after reflection or refraction, then this point is called image of the object.

(vi) Real Image The image obtained by the real meeting of light rays, is called a real image.

Real image can be obtained on a screen. Real image is inverted.

(vii) Virtual Image The image obtained when light rays are not really meeting but appears to meet only, is called a virtual image.

Reflection of Light
The rebouncing back of light rays into the same medium on striking a highly polished surface such as a mirror, is called reflection of light.

Laws of Reflection

There are two laws of reflection.

(i) The incident ray, the reflected ray and the normal at the point of incidence all three lie in the same plane.

(ii) The angle of incidence (i) is always equal to the angle of reflection (r).

Types of Reflection

(i) Regular Reflection When a parallel beam of reflected light rays is obtained for a parallel beam of incident light rays after reflection from a plane reflecting reflection is called regular reflection.

(ii) Irregular or Diffused Reflection When a non-parallel beam of reflected light rays is obtained for a parallel beam of incident light rays after reflection from a surface, then such type of reflection is called irregular or diffused reflection.

Mirror

A smooth and highly polished reflecting surface is called a mirror.

(i) Plane Mirror A highly polished plane surface is called a plane mirror.

Different properties of image formed by plane mirror

Size of image = Size of object

Magnification == Unity

Distance of image == Distance of object

A plane mirror may form a virtual as well as real image.

A man may see his full image in a mirror of half height of man.

When two plane mirror are held at an angle θ, the number of images of an object placed between them is given as below

(a) n = [(360° / θ) – 1 ], where 360° / θ is an integer.

(b) n = integral part of 360° / θ, when 360° is not an integer.

[A plane mirror may form a real image, when the pencil of light incident on the mirror is convergent. Children, during their play form an image of sun as wall by a strip of plane mirror.]

Kaleidoscope and periscope employ the principle of image formation by plane mirror.

If keeping an object fixed a plane mirror is rotated in its plane by an angle θ, then the reflected ray rotates in the same direction by an angle 2 θ.

Focal length as well as radius of curvature of a plane mirror is infinity. Power of a plane mirror is zero.

An image formed by a plane mirror is virtual, erect, laterally inverted, of same size as that of object and at the same distance as the object from the mirror.

(ii) Spherical Mirror A highly polished curved surface whose reflecting surface is a cut part of a hollows at glass sphere is called a spherical mirror. Spherical mirrors are of two types

(a) Concave Mirror A spherical mirror whose bent in surface is reflecting surface, is called a concave mirror.

(b) Convex Mirror A spherical mirror whose bulging out surface is reflecting surface, is called a convex mirror.

Some Terms Related to Spherical Mirrors are Given Below

(i) Centre of Curvature It is the centre of the sphere of which the mirror or lens is a part.

(ii) Radius of Curvature (R) The radius of the hollow sphere of which the mirror is a part, is called radius of curvature.

(iii) Pole The central point of the spherical mirror is called its pole (P).

(iv) Focus When a parallel beam of light rays is incident on a spherical mirror, then after reflection it meets or appears to meet at a point on principal axis, which is called focus of the spherical mirror.

(v) Focal Length The distance between the pole and focus is called focal length (f).

Relation between focal lengthand radius of curvature is given by

f=R/2

The power of a mirror is given as P = 1/f (metre)

(vi) Mirror formula 1/f  = 1/v + 1/u

.

Linear Magnification

The ratio of height of image (1) formedby a mirror to the height of the object (O) is called linear magnification (m).

Lin ear magnification (m) = I/O = -v/u

Areal and Axial Magnification

The ratio of area of image to the area of object is called areal magnification.

Sign Convention for Spherical Mirrors

1. All distances are measured from the pole of the mirror.
2. Distances measured in the direction of incident light rays are taken as positive.
3. Distances measured in opposite direction to the incident light rays are taken as negative.
4. Distances measured above the principal axis are positive.
5. Distances measured below the principal axis are negative.

Lateral lnversion

In the image formed by a plane mirror the right side of the object ppears as left side and vise-versa. This phenomena is called lateral inversion.

When object is placed between pole and focus of a concave mirror, then its virtual, erect and magnified image is formed.

A convex mirror forms a virtual, erect and diminished image for all conditions of object.

The focal length of concave mirror is taken negative and for a convex mirror taken as positive.

Refraction of Light

The deviation of light rays from its path when it travells from one transparent medium to another transparent medium, is called refraction of light.

Cause of Refraction

The speed of light is different i.n different media.

Laws of Refraction

(i) The incident ray, the refracted ray and the normal at the point of incidence, all three lies in the same plane.

(ii) The ratio of sine of angle of incidence to the sine of angle of refraction is constant for a pair of two media,

where ₁&mu₂ is called refractive index of second medium with respect to first medium.

This law is also called Snell’s law.

Refractive Index

The ratio of speed of light in vacuum (c) to the speed of light in any medium (u) is called refractive index of the medium.

Refractive index of a medium,

μ = c/v

Refractive index of water =4/3 = 1.33; Refractive index of glass = 3/2 = 1.50

When light is reflected by a denser medium, phase difference of π radian or path difference of λ/2 or time difference T/2 is produced. This is known as Stoke’s law. Distance x travelled by light in a medium of refractive index μ is equal to distance (&mux) travelled in vacuum.

Time taken by light to traverse a thickness x of time =μx/c, where c = velocity of light in vacuum.

Relative Refractive Index

The refractive index of second medium with respect to first medium

Cauchy’s Formula

Critical Angle

The angle of incidence in a denser medium for which the angle of refraction in rarer medium becomes 90°. is called critical angle (C).

Critical angle for diamond = 24°

Critical angle for glass = 42° :

Critical angle for water = 48° :

Refractive index of denser medium μ = 1/sin C

Total Internal Reflection (TIR)

When a light ray travelling from a denser medium towards a rarer medium is incident at the interface at an angle of incidence greater than critical angle, then
light rays reflected back in to the denser medium. This phenomena is called TIR.

Critical angle increases with temperature.

The refractive index is maximum for violet colour of light and minimum for red colour of light. i.e., μ_v > μ _R therefore critical angle is maximum for red colour of light and minimum for violet colour of light, i.e., C_v < C_R

Total internal reflection occurs if angle of incidence in denser medium exceeds critical angle.

Mirage is an optical illusion observed in deserts and roads on a hot day when the air near the ground is holler and hence rarer than the air above.

Optical Fibres

are also based on the phenomenon of total internal reflection. Optical fibres consist of several thousands of very long fine quality fibres of glass or quartz. The diameter of each fibre is of the order of 10-4 cm with refractive index of material being of the order of 1.5. Optical fibres are used in transmission and reception of electrical signals by converting them first into light signals.
For Refraction at a Convex or Concave Spherical Surface

where, μ= refractive index, u = distance of object, v = distance of image and R = radius of curvature of the spherical surface

Lens

A lens is a uniform transparent medium bounded between two spherical or one spherical and one plane surface.

Convex Lens

A lens which is thinner at edges and thicker at middle is called a convex or converging lens.

Concave Lens

A lens which is thicker at edges and thinner at middle, is called a concave or diverging lens.

Lens Formula
1/f = 1/v – 1/u
where, f = focal length of the lens, U = distance of object, U = distance of image.

Lens Maker’s formula
1/f=(μ – 1) (1/R₁ – 1/R₂)

where, μ = refractive index of the material of the lens and R₁ and R₂ are radii of curvature of the lens.

Power of a Lens

The reciprocal of the focal length of a lens, when it is measured in metre, is called power of a lens.

Power of a lens, (P)= 1/f(metre)

Its unit is diopter (D).

The power of a convex (converging) lens is positive and for a concave (diverging) lens it is negative.

Focal Length of a Lens Combination

(i) When lenses are in contact 1/F – 1/f₁ + 1/f₂

Power of the combination P = P₁ + P₂

(ii) When lenses are separated by a distance d

1/F = 1/f₁ + 1/f₂ – d/f₁f₁

Power of the combination
P = P₁ + P₂ – dP₁P₂

Linear Magnification

m = I/O = v/u

For a small sized object placed linearly along the principal axis, its axial (longitudinal) magnification is given by

Axial magnification = – dv/du = (v/u)²
=(f/f+u)² = (f-v/f)²

Focal Length of a Convex Lens by Displacement Method

Focal length of the convex lens f = (a² – d²) / 4a
where, a = distance between the image pin and object pin and
d = distance between two positions of lens.

The distance between the two pins should be greater than four times the focal length of the convex lens, i.e., a > 4f.

Height of the object O = √I₁I₂

Cutting of a Lens

(i) If a symmetrical convex lens of focal length f is cut into two parts along its optic axis, then focal length of each part (a plano convex lens) is 2f. However, if the two parts are joined as shown in figure, the focal length of combination is again f.

(ii) If a symmetrical convex lens of focal length f is cut into two parts along the principal axis, then focal length of each part remains unchanged as f. If these two parts are joined with curved ends an one side, focal length of the combination is f/2. But on joining two 2 parts in opposite sense the net focal length becomes

Prism

Prism is uniform transparent medium bounded between two refracting surfaces, inclined at an angle.

Angle of Deviation

The angle sub tended between the direction of incident light ray and emergent light fay from a prism is called angle of deviation (δ).

Prism Formula

The refractive index of material of prism

Dispersion of Light

The splitting of white light into its constituent colours in the sequence ofVIBGYOR, on passing through a prism. is called dispersion of light.

The refractive index μ_v > μ_R therefore violet colour deviates most and red colour deviates least. i.e., δ_v > δ_R.

Angular Dispersion

The angle suhtended between the direction of emergent violet and red rays of light from a prism is called angular dispersion.

Angular dispersion

(θ) δ_v – δ_R = (μ_v – μ_R A

where δ_v and δ_R are angle of deviation.

Dispersive Power

W = θ/δ_Y = (μ_v – μ _R) / (μ_Y – 1)

where μ_Y = (μ _v + μ _R ) / 2, is mean refractive index.

Human Eye

Human eye is an optical instrument which forms real image of the objects on retina.

Retina colours contains lakhs of cone and rod cells which of light and intensities of light respectively.

Ciliary muscles change the focal length of eye lens. This power of eye is called power of accomadation of eye.

Different defects of vision of human eye are described below

(i) Myopia or Short-Sightedness It is a defect of eye due to which a person can see near by objects clearly but cannot see far away objects clearly.

In this defect, the far point of eye shifts from infinity to a nearer distance.

This defect can be removed by using a concave lens of appropriate power.

(ii) Hypermetropia or Long-Sightedness In this defect, a person can see far away objects clearly but cannot see near by objects clearly.

In this defect the near point of eye shifts away from the eye.

This defect can be removed by using a convex lens of appropriate power.

(iii) Astigmatism In this defect, a person cannot focus on horizontal and vertical lines at the same distance at the same time.

This defect can be removed by using suitable cylindrical lenses.

(iv) Colour Blindness In this defect, distinguish between few colours. a person is unable to The reason of this defect is the absence few colours. of cone cells sensitive for

This defect cannot be removed.

(v) Cataract In this defect. an opaque white membrane is developed on cornea due to which person lost power of vision partially on completely.

This defect can be removed by removing this membrane through surgery.

Camera

A photograph camera consists of a light proof box, at one end of which a converging lens system is fitted. A light sensitive film is fixed at the other end of the box, opposite to the lens system. A real inverted image of the object is formed on the film by the lens system.

f-Number for a Camera: The f-number represent the size of the aperture.

f-number =Focal length of the lens (F) / Diameter of the lens(d)

Generally 2, 2.8, 4, 5.6, 8, 11, 22, 32 are f-numbers.

The amount of light (L) entering the camera is directly proportional to the area (A) of the aperture, i.e.,

L ∝A∝ d²

Brightness of Image ∝ (d2/f2)

where, d = dia meter of the lens and F = focal length of the lens.

Exposure time is the time for which light is incident of photographic film.

Simple Microscope

It is used for observing magnified images of objects. It is consists of a converging lens of small focal length.

Magnifying Power

(i) When final image is formed at least distance of distinct vision (D), then M=1+d/f

where, f= focal length of the lens.

(ii) When final image is formed at infinity, then M = D/f

Compound Microscope

It is a combination of two convex lenses called objective lens and eye piece separated by a distance. Both lenses are of small focal lengths but f_o < f_e, where f_o and f_e are focal lengths of objective lens and eye piece respectively

Magnifying Power

M = v_o / u_o {1 + (D/f_o)

Where v_o= distance of image, formed by objective lens and
u_o = distance of object from the objective

(ii) When final image is formed at infinity, then
M = v_o/u_o . D/f_e

Astronomical Telescope

It is also a combination of two lenses, called objective lens and eye piece, separated by a distance. It is used for observing distinct images of heavenly bodies like stars, planets etc.

Magnifying Power

(i) When final image is formed at least distance of distinct vision (D), then M = f_o/f_e {1+ (D/f_e)} where f_o and f_e are focal lengths of objective and eyepiece respectively.

Length of the telescope (L) = (f_o + u_e)

where, u_e = distance of object from the eyepiece.

(ii) When final image is formed at infinity, then M = f_o/f_e

Length of the telescope (L) = f_o + f_e

For large magnifying power of a telescope f_o should be large and f_e should be small.

For large magnifying power of a microscope; f_o < f_e should be small.

Resolving Power

The ability of an optical instrument to produce separate and clear images of two near by objects, is called its resolving power.

Limit of Resolution

The minimum distance between two near by objects which can be just resolved by the instrument, is called its limit of resolution (d).

Resolving power of a microscope = 1/d = 2 μ sin θ / λ

where, d = limit of resolution, λ = wavelength of light used.
μ = refractive index of the medium between the objects and objective lens and θ = half of the cone angle.

Resolving power of a telescope = 1/dθ = d/1.22 λ

where, dθ = limit of resolution, A = wavelength of light used and
d = diameter of aperture of objective

Aberration of Lenses

The image formed by the lens suffer from following two rnaiIl drawbacks

(i) Spberical Aberration Aberration of the lens due to which the rays passes through the lens are not focussed at a single and the image of a point object placed on the axis is blurred. called spherical aberration.

It can be reduced by using

• lens of large focal lengths
• plano-convex lenses
• crossed lenses
• combining convex and concave lens

(ii) Chromatic Aberration Image of a white object formed by lens is usually coloured and blurred. This defect of the image produced by lens is called chromatic aberration.

Scattering of Light

When light passes through a medium in which particles are suspended whose size is of the order of wavelength of light, then light on striking these particles, deviated in different directions. These phenomena is called scattering of light.

According to the Lord Rayleigh, the intensity of scattered light

I ∝ 1/λ⁴

Therefore, red colour of light is scattered least and violet colour of light is scattered most.

Daily Life Examples of Scattering of Light

1. Blue colour of sky.
2. Red colour of signals of danger. /
3. Black colour of sky in the absence of atmosp,here
4. Red colour of the time of sun rise and sun set.
5. The human eye is most sentitive to yellow colour

An electric current which vary for a small finite time, while growing from zero to maximum or decaying from maximum to zero, is called a transient current.

Growth of Current in an Inductor

Growth of current in an inductor at any instant of time t is given by

I = I_o(1 – e ^{-Rt / L})

where, I_o = maximum current, L = self inductance of the inductor and R = resistance of the circuit.

Here R / L = τ, is called time constant of a L – R circuit.

Time constant of a L – R circuit is the time in which current in the circuit grows to 63.2% of the maximum value of current.

Decay of current in an inductor at any time t is given by

I = I_oe ^{-Rt / L}

Time constant of a L – R circuit is the time in which current decays to 36.8% of the maximum value of current.

Charging and Discharging of a Capacitor

The instantaneous charge on a capacitor on charging at any instant of time t is given by

q = q_o(1 – e ^{– t / RC})

where RC = τ, is called time constant of a R – C circuit.

The instantaneous charge on a capacitor in discharging at any instant of time t is given by q = q_oe ^{– t / RC}

Time constant of a R – C circuit is the time in which charge in the capacitor grows to 63.8% or decay to 36.8% of the maximum charge on capacitor.

Alternating Current

An electric current whose magnitude changes continuously with time and changes its direction periodically, is called an alternating current.

The instantaneous value of alternating current at any instant of time t is given by

I = I_o sin ωt

where, 10 = peak value of alternating current.

The variation of alternating current with time is shown in graph given below

Mean or average value of alternating current for first half cycle

I_m = 2I_o / π = 0.637 I_o

Mean or average value of alternating current for next half cycle

I’_m = – 2I_o / π = – 0.637 I_o

Mean or average value of alternating current for one complete cycle = O.

Root mean square value of alternating current

I_v = I_rms = I_o / √2 = 0.707 I_o

Where, I_o = peak value of alternating current.

Root mean square value of alternating voltage

V_rms = V_o / √2 = 0.707 I_o = 0.707 V_o

Reactance

The opposition offered by an inductor or by a capacitor in the path of flow of alternating current is called reactance.

Reactance is of two types

(i) Inductive Reactance (X_L) Inductive reactance is the resistance offered by an inductor.

Inductive reactance (X_L) = Lω = L2πf = L2π / T

Its unit is ohm. X_L ∝ f

For direct current, X_L = 0 (f = 0)

(ii) Capacitive Reactance (X_c) Capacitive reactance is the resistance offered by an inductor

Capacitive reactance,

X_c = 1 / Cω = 1 / C2πf = T / C 2π

Its unit is ohm X_c ∝ 1 / f

For direct current, X_c = ∞ (f = 0)

Impedance

The opposition offered by an AC circuit containing more than one out of three components L, C and R, is called impedance (Z) of the circuit.

Impedance of an AC circuit, Z = √R² + (X_L – X_C)²

Its SI unit is ohm.

Power in an AC Circuit

The power is defined as the rate at which work is being in the circuit.

The average power in an AC circuit,

P_av = V_rms i_rms cos θ

= V / √2 i / √2 cos θ = Vi / √2 cos θ

where, cos θ = Resistance(R) / Impedance (Z) is called the power factor 0f AC circuit.

Current and Potential Relations

Here, we will discuss current and potential relations for different AC circuits.

(i) Pure Resistive Circuit (R circuit)

(a) Alternating emf, E = E_o sin ωt

(b) Alternating current, I = I_o sin ωt

(c) Alternating emf and alternating current both are in the same phase.

(d) Average power decay, (P) = E_v . I_v

(e) Power factor, cos θ = 1

(ii) Pure Inductive Circuit (L Circuit)

(a) Alternating emf, E = E_o sin ωt

(b) Alternating current, I = I_o sin (ωt – π / 2)

(c) Alternating current lags behind alternating emf by π / 2.

(d) Inductive reactance, X_L = Lω = L2πf

(e) Average power decay, (P) = 0

(f) Power factor, cos θ = cos 90° = 0

(iii) Pure Capacitive Circuit

(a) Alternating emf, E = E_o sin ωt

(b) Alternating current, I = I_o sin (ωt + π / 2)

(c) Alternating current lags behind alternating emf by π / 2.

(d) Inductive reactance, X_L = Cω = C2πf

(e) Average power decay, (P) = 0

(f) Power factor, cos θ = cos 90° = 0

(iv) R – C Circuit

E = E_o sin ωt

I = E_o / 2 sin (ωt – φ)

Z = √R² + (1 / ωC)²

tan φ = – 1 / ωC / R

Current leading the voltage by φ

V² = V²_R = V²_C

(v) L – C Circuit

(vi) L – C – R Circuit

(a) Alternating emf, E = E_o sin Ωt

(b) Alternating current, I = I_o sin (Ωt ± θ)

(c) Alternating current lags leads behind alternating emf by ω.

(d) Resultant voltage, V = √V²_R + (V_L – V_C)²

(e) Impedance, Z = √R² + (X_L – X_C)²

(f) Power factor, cos θ = R / Z = R / √√R² + (X_L – X_C)²

(g) Average power decay, (P)= E_VI_V cos θ

Resonance in AC Circuit

The condition in which current is maximum or impedance is minimum in an AC circuit, is called resonance.

(i) Series Resonance Circuit

In this circuit components L, C and R are connected in series.

At resonance = X_L = X_C

Resonance frequency f = 1 / 2π√LC

A series resonance circuit is also known as acception circuit.

(ii) Parallel Resonance Circuit

In this circuit L and C are connected in parallel with each other.

At resonance, X_L = X_C

Impedance (Z) of the circuit is maximum.

Current in the circuit is minimum.

Wattless Current

Average power is given by

P_av = E_rms = I_rms cos θ

Here the I_rms cos φ contributes for power dissipation. Therefore, it is called wattless current.

AC Generator or Dynamo

It is a device which converts mechanical energy into alternating current energy.

Its working is based on electromagnetic induction.

The induced emf produced by the AC generator is given by

e = NBAω sin ωt = e_o = sin ωt

There are four main parts of an AC generator

(i) Armature It is rectangular coil of insulated copper wire having a large number of turns.

(ii) Field Magnets These are two pole pieces of a strong electromagnet.

(iii) Slip Rings These are two hollow metallic rings.

(iv) Brushes These are two flexible metals or carbon rods, which remains slightly in contact with slip rings .

Note An DC generator or dynamo contains split rings or commutator inspite of slip rings.

DC Motor

It is a device which converts electrical energy into mechanical energy.

Its working is based on the fact that when a current carrying coil is placed in uniform magnetic field a torque acts on it.

Torque acting on a current carrying coil placed in uniform magnetic field

τ = NBIA sin θ

When armature coil rotates a back emf is produced in the coil.

Efficiency of a motor,

η = Back emf / Applied emf = E / V

Transformer

It is a device which can change a low voltage of high current into a high voltage of low current and vice-versa.

Its working is based on mutual induction.

There are two types of transformers.

(i) Step-up Transformers It converts a low voltage of high current into a high voltage of low current.

In this transformer,

N_s > N_P, E_s > E_P

and I_P > I_S

(ii) Step-down Transformer It converts a high voltage of low current into a low voltage of high current.

In this transformer,

N_P > N_S, E_P > E_S and I_P < I_S

Transformation Ratio

Transformation ratio,

K = N_S / N_P = E_S / E_P = I_P / I_S

For step-up transformer, K > 1

For step-down transformer, K < 1

Energy Losses in a Transformer

The main energy losses in a transformer are given below

1. Iron loss
2. Copper loss
3. Flux loss
4. Hysteresis loss
5. Humming loss

Important Points

• Transformer does not operate on direct current. It operates only on alternating voltages at input as well as at output.
• Transformer does not amplify power as vacuum tube.
• Transformer, a device based on mutual induction converts magnetic energy into electrical energy.
• Efficiency, η = Output power / Input power

Generally efficiency ranges from 70% to 90%.

• A choke coil is a pure inductor. Average power consumed per cycle is zero in a choke coil.
• A DC motor connects DC energy from a battery into mechanical energy of rotation.
• An AC dynamo/generator produces are energy from mechanical energy of rotation of a coil.
• An induction coil generates high voltages of the order of 1OS V from a battery.

It is based on the phenomenon of mutual induction.

Faraday’s Laws of Electromagnetic Induction

(i) Whenever the magnetic flux linked with a circuit changes, an induced emf is produced in it.

(ii) The induced emf lasts so long as the change in magnetic flux continues.

(iii) The magnitude of induced emf is directly proportional to the rate of change in magnetic flux, i.e.,

E ∝ dφ / dt ⇒ E = – dφ / dt

where constant of proportionality is one and negative sign indicates Lenz’s law.

Here, flux = NBA cos θ, SI unit of φ = weber,

CGS unit of φ = maxwell, 1 weber = 10⁸ maxwell,

Dimensional formula of magnetic flux

[φ] = [ML²T^-2A^-2]

Lenz’s law

The direction of induced emf or induced current is always in such a way that it opposes the cause due to which it is produced.

Lenz’s law is in accordance with the conservation of energy.

Note To apply Lenz’s law, you can remember RIN or ® In (when the loop lies on the plane of paper)

(i) RIN In RIN, R stands for right, I stands for increasing and N for north pole (anticlockwise). It means, if a loop is placed on the right side of a straight current carrying conductor and the current in the conductor is increasing, then induced current in the loop is anticlockwise

(ii) ®IN In ® IN suppose the magnetic field in the loop is perpendicular to paper inwards ® and this field is increasing, then induced current in the loop is anticlockwise

Motional Emf

If a rod of length 1 moves perpendicular to a magnetic field B, with a velocity v, then induced emf produced in it given by

E = B * v * I = bvl

If a metallic rod of length 1 rotates about one of its ends in a plane perpendicular to the magnetic field, then the induced emf produced across its ends is given by

E = 1 / 2 bωr² = BAf

where, ω = angular velocity of rotation, f = frequency of rotation and A = πr² = area of disc.

The direction of induced current in any conductor can be obtained from Fleming’s right hand rule.

A rectangular coil moves linearly in a field when coil moves with constant velocity in a uniform magnetic field, flux and induced emf will be zero.

A rod moves at an angle θ with the direction of magnetic field, velocity E = – Blv sin θ.

An emf is induced

(i) When a magnet is moved with respect to a coil.

(ii) When a conductor falls freely in East-West direction.

(iii) When an aeroplane flies horizontally.

(iv) When strength of current flowing in a coil is increased or decreased, induced current is developed in the coil in same or opposite direction.

(v) When a train moves horizontally in any direction.

Fleming’s Right Hand Rule

If we stretch the thumb, the forefinger and the central finger of right hand in such a way that all three are perpendicular to each other, th. if thumb represent the direction of motion, the forefinger represent tile direction of magnetic field, then centra} finger will represent the
direction of induced current.

If R is the electrical resistance of the circuit, then induced current in the circuit is given by I = E / R

If induced current is produced in a coil rotated in uniform magnetic field, then

I = NBA ω sin ωt / R = I_o sin ωt

where, I_o = NBA ω = peak value of induced current,

N = number of turns in the coil ,

B= magnetic induction,

ω = angular velocity of rotation and

A = area of cross-section of the coil.

Eddy Currents

If a piece of metal is placed in a varying magnetic field or rotated high speed in a uniform magnetic field, then induced current set up
the piece are like whire pool of air, called eddy currents.

The magnitude 0f eddy currents is given by i = – e / R = dφ / dt / R, where R is the resistance.

Eddy currents are also known as Facault’s current.

Self-Induction

The phenomena of production of induced emf in a circuit due to change in current flowing in its own, is called self induction.

Coefficient of Self-Induction

The magnetic flux linked with a coil

φ = LI

where, L = = coefficient of self induction.

The induced emf in the coil

E = – L dl / dt

it unit of self induction is henry (H) and its dimensional formula is [ML²T ^-2A^-2].

Self – inductance of a long solenoid is given by normal text

L = μ_o N² A / l = μ_o n² Al

where. N = total number of turns in the solenoid,

1 = length of the coil, n = number of turns in the coil and

A = area of cross-section of the coil.

If core of the solenoid is of any other magnetic material, then

L = μ_o μ_r N² A / l

Self – inductance of a toroid L = μ_o N² A / 2πr

Where, r = radius of the toroid

Energy stored in an inductor E = 1 / 2 LI²

Mutual Induction

The phenomena of production of induced emf in a circuit due to the change in magnetic flux in its neighbouring circuit, is called mutual induction.

Coefficient of Mutual Induction

If two coils are coupled with each, other then magnetic flux linked with a Coil (secondary coil)

φ = MI

where M is coefficient of mutual induction and I is current flow in through primary coil.

The induced emf in the secondary coil

E = – M dl / dt

where dl / dt is the rate of change of current through primary coil.

The unit of coefficient of mutual induction is henry (H) and its dimension is [ML²T^-2A^-2].

The coefficient of mutual induction depends on geometry of two coils, distance between them and orientation of the two coils.

Coefficient of Coupling

Two coils are said to be coupled if full a part of the fuse produced by one links with the other.

K = √M / L₁ L2, where L₁ and L₂ are coefficients of self-induction of the two coils and M is coefficient of mutual induction of the two coils.

Coefficient of coupling is maximum (K = 1) in case (a), when coils are coaxial and minimum in case (b), when coils are placed a right angles.

Mutual inductance of two long coaxial solenoids is given by

M = μ N₁ N₂ A / l

= μ n₁ n₂ Al

where N₁ and N₂ are total number of turns in both coils, n₁ n₂ are number of turns per unit length in coils, A is area of cross-section of coils and 1 is length of the coils.

Grouping of Coils

(a) When three coils of inductances L₁, L₂ and L₃ are connected in series and the coefficient of coupling K = 0, as in series, then

L = L₁ + L₂ + L₃

(b) When three coils of inductances L₁, L₂ and L₃ are connected in parallel and the coefficient of coupling K= 0 as in parallel, then

L = 1 / L₁ + 1 / L₂ + 1 / L₃

If coefficient of coupling K = 1, then

(i) In series

(a) If current in two coils are in the same direction, then

L = L₁ + L₂ + 2M

(b) If current in two coils are in opposite directions, then

L = L₁ + L₂ – 2M

(ii) In parallel

(a) If current in two coils are in same direction, then

L = L₁ L₂ – M² / L₁ + L₂ + 2M

(b) If current in two coils are in opposite directions, then

L = L₁ L₂ – M² / L₁ + L₂ – 2M

Natural Magnet

A natural magnet is an ore of iron (Fe₃O₄), which attracts small pieces of iron, cobalt and nickel towards it.

Magnetite or lode stone is a natural magnet.

Artificial Magnet

A magnet which is prepared artificially is called an artificial magnet, e.g., a bar magnet, an electromagnet, a magnetic needle, a horse-shoe magnet etc.

According to molecular theory, every molecular of magnetic substance (whether magnetised or not) is a complete magnet itself.

The poles of a magnet are the two points near but within the ends of the magnet, at which the entire magnetism can be assumed to be concentrated.

The poles always occur in pairs and they are of equal strength. Like poles repel and unlike poles attract.

Properties of Magnet

(i) A freely suspended magnet always aligns itself into north-south direction.

(ii) Like magnetic poles repel and unlike magnetic poles attract each other.

(iii) Magnetic poles exist in pair.

Coulomb’s Law

The force of interaction acting between two magnetic poles is directly proportional to the product of their pole strengths and inversely proportional to the square of the distance between them.

F = μ_o / 4π . m₁m₂ / r²

where m₁, m₂ = pole strengths, r = distance between poles and μ_o = permeability of free space.

Magnetic Dipole

Magnetic dipole is an arrangement of two unlike magnetic poles of equal pole strength separated by a very small distance, e.g., a small bar magnet, a magnetic needle, a current carrying loop etc.

Magnetic Dipole Moment

The product of the distance (2 l) between the two poles and the pole strength of either pole is called magnetic dipole moment.

Magnetic dipole moment

M = m (2 l)

Its SI unit is ‘joule/tesla’ or ‘ampere-metre²‘.

Its direction is from south pole towards north pole.

Magnetic Field Due to a Magnetic Dipole

(1) On Axial Line

If r > > l, then

B = μ_o / 4 π 2M / r³

(ii) On Equatorial Line

B = μ_o / 4 π M / (r² + l²)^{3 / 2}

If r > > l, then

B = μ_o / 4 π 2M / r³

Torque Acting on a Magnetic Dipole

When a Magnetic Dipole (M) is placed in a uniform magnetic field (B), then a Torque acts on it, Which is given by

τ = M * B

or τ = MB sin θ

Where θ is angle between the dipole axis and magnetic field.

Potential Energy of a Magnetic Dipole in a Uniform Magnetic Field

The work done in rotating the dipole against the action of the torque is stored as potential energy of the dipole.

Potential Energy, U = W = – MB cos θ = – M . B

Current Carrying Loop

A current carrying loop behaves as a magnetic dipole. If we look the upper face of the loop and current is flowing anti-clockwise,then it has a north polarity and if current is flowing clockwise.then it has a south polarity.

Magnetic dipole moment of a current carrying loop is given by

M = IA

For N such turns M = NIA

Where I = current and A = area of cross-section of the coil.

Gauss’s Law in Magnetism

Surface integral of magnetic field over any closed or open surface is always m.

This law tells that the net magnetic flux through any surface is always zero.

[When in an atom any electron revolve in an orbit it is equivalent to a current loop. Therefore, atom behaves as a magnetic dipole].

Magnetic Moment of an Atom

Magnetic moment of an atom M = 1 / 2 eωr²

where e = charge on an electron, ω = angular velocity of electron and r = radius of orbit.

or M = n eh / 4πm

where h = Planck’s constant and m ~ mass of an electron and eh / 4πm = μ_B, called Bohr magneton and its value is 9.27 * 10^-24 A-m².

Earth’s Magnetism

Earth is a huge magnet. There are three components of earth’s magnetism

(i) Magnetic Declination (θ) The smaller angle subtended between the magnetic meridian and geographic meridian is called magnetic declination.

(ii) Magnetic Inclination or Magnetic Dip (δ) The smaller angle sub tended between the magnetic axis and horizontal is called magnetic inclination on magnetic dip.

(iii) Horizontal Component of Earth’s Magnetic Field (H) If B is the intensity of earth’s magnetic field then horizontal component of earth’s magnetic field H = B cos δ

It acts from south to north direction.

Vertical component of earth’s magnetic field

V = B sin δ

∴ B = √H²+ V²

and tan δ = V / H

Angle of dip is zero at magnetic equator and 90° at poles.

Magnetic Meridian

A vertical plane passing through the magnetic axis is called magnetic mendian.

Geographic Meridian

A vertical plane passing through the geographic axis is called geographic meridian.

Magnetic Map

Magnetic map is obtained by drawing lines on the surface of earth. which passes through different places having same magnetic elements.

The main lines drawn on earth’s surface are g1 en below

(i) Isogonic Line A line joining places of equal declination is called on isogonic line.

(ii) Agonic Line A line joining places of zero declination is called an agonic line

(iii) Isoclinic Line A line joining places of equal inclination or dip is called an aclinic line,

(iv) Aclinic Line A hne joining places of zero inclination or dip is called an aclinic line.

(v) Isodynamic Line A line joining places of equal horizontal component of earth’s magnetic field (H) is called an isodynamic line.

Magnetic Latitude

(i) If at any place, the angle of dip is δ and magnetic latitude is λ then tan δ = 2 tan λ

(ii) The total Intensity of earth’s magnetic field

I = I₀ √1 + 3 sin² λ

where I_o = M / R³

It is assumed that a bar magnet of earth has magnetic moment M and radius of earth is R.

(Magnetic maps are maps obtained by drawing lines passing through different places on the surface of earth, having the same value of a magnetic element.)

Neutral Points

Neutral point of a bar magnet is a point at which the resultant magnetic field of a bar magnet and horizontal component of earth’s magnetic field are zero.

When north pole of a bar magnet is placed towards south pole of the earth. then neutral point is obtained on axial line.

B = μ_o / 4π 2Mr / (r² – l²)² = H

If r > > l, then B = μ_o / 4π 2M / r³ = H

When north pole of a bar magnet is placed towards north pole of the earth, then neutral point is obtained on equatorial line

B = μ_o / 4π 2Mr / (r² + l²)² = H

If r > > l, then B = μ_o / 4π 2M / r³ = H

Tangent Law

When a bar magnet is freely suspended under the combined effect of two uniform magnetic fields of intensities B and H acting at 90° to each other, then it bar magnet comes to rest making an angle 0 with the direction of H, then

B = H tan θ

Deflection Magnetometer

It is a device used to determine M and H. Its working is based on tangent law.

Deflection magnetometer can be used into two settings

(i) Tangent A setting In this setting the arms of the magnetometer are along east-west and magnet is parallel to the arms.

In equilibrium

B = H tan θ

μ_o / 4π 2M / d³ = H tan θ

(ii) Tangent B setting In this setting the arms of the magnetometer are along north-south and magnet is perpendicular to these arm in equilibrium

μ_o / 4π M / d³ = H tan θ

In above setting the experiment can be performed in two ways.

(a) Deflection method In this method one magnet is used at a time and deflection in galvanometer is observed. Ratio of magnetic dipole moments of the magnets

M₁ / M₂ = tanθ₁ / tanθ₂

where θ₁ and θ₂ are mean values of deflection for two magnets.

(b) Null method In this method both magnets are used at a time and no deflection condition is obtained. If Magnets are at distance d₁ and d₂ then

M₁ / M₂ = (d₁ / d₂)³

Tangent Galvanometer

It is a device used for detection and measurement of low electric currents. Its working is based on tangent law. If θ is the deflection produced in galvanometer when I current flows through it, then

I = 2R / Nμ_o H tan θ = H / G tan θ = K tan θ

Where G = Nμ / 2R is called galvanometer constant and K = H / G, is called reduction factor of tangent galvanometer.

Here N is number of turns in the coil and R is radius of the coil.

Tangent galvanometer is also called moving magnet type galvanometer.

Vibration Magnetometer

It is based on simple harmonic oscillations of a magnet suspended in uniform magnetic field.

Time period of vibrations is given by

T = 2π √I / MH

where, I = moment of inertia of the magnet,

M = magnetic dipole moment of the magnet and

H = horizontal component of earth’s magnetic field.

When two magnets of unequal size are placed one above the other and north poles of both magnets are towards geographic north then time period of oscillations is given by

T₁ = 2π√I₁ + I₂ / (M₁ + M₂) H

If north pole of first magnet and south pole of second magnet is towards geographic north, then time period of oscillations is given by

T₂ = 2π√(I₁ + I₂) / (M₁ – M₂) H

Then, M₁ / M₂ = T²₂ + T²₁ / T²₂ – T²₁

Magnetic Flux

The number of magnetic lines of force passing through any surface is called magnetic flux linked with that surface.

Magnetic flux (phi;) = B . A = BA cos θ

where B is magnetic field intensity or magnetic induction, A is area of the surface.

Its unit is ‘weber’.

Magnetic Induction

The magnetic flux passing through per unit normal area, is called magnetic induction.

Magnetic induction (B) = φ / A

Its unit is ‘waber/metre²‘ or ‘tesla’.

Magnetic of Material

To describe the magnetic properties of materials, following terms are required

(i) Magnetic Permeability It is the ability of a material to permit the passage of magnetic lines of force through it.

Magnetic permeability (μ) = B / H

where B is magnetic induction and H is magnetising force or magnetic intensity.

(ii) Magnetising Force or Magnetic Intensity The degree up to which a magnetic field can magnetise a material is defined in terms of magnetic intensity.

Magnetic intensity (H) = B / μ

(iii) Intensity of Magnetisation The magnetic dipole moment developed per unit volume of the material is called intensity of magnetisation.

Intensity of magnetisation (I) = M / V = m / A

where V = volume and A = area of cross-section of the specimen.

Magnetic induction B = μ_o (H + I)

(iv) Magnetic Susceptibility(χm) The ratio of the intensity of magnetisation (1) induced in the material to the magnetising force (H) applied, is called magnetic susceptibility.

Magnetic Susceptibility(χm) = I / H

[Relation between Magnetic Permeability and Susceptibility is given by

μ = μ_o (1 + χm) ]

Classification of Magnetic Materials

On the basis of their magnetic properties magnetic materials are divided into three categories

(i) Diamagnetic substances

(ii) Paramagnetic substances

iii) Ferromagnetic substances

The atoms of a paramagnetic substance contain even number of electrons.

The atoms or molecules of a paramagnetic substance do not possess any net magnetic moment.

The atoms of a paramagnetic material contain odd number of electrons.

Every atom or molecule of a paramagnetic substance has its own magnet moment, i.e., its each atom or molecule is a tiny magnet.

In a ferromagnetic substance, there are several tiny regions called domains. Each domain contain approximately 1010 atoms.

Each domain is a strong magnet as all atoms or molecules in a domain have same direction of magnetic moment.

Curie Law in Magnetism

The magnetic susceptibility of a paramagnetic substance is inversely proportional to its absolute temperature.

χm ∝ 1 / T ⇒ χm T = constant

where χm = magnetic susceptibility of a para magnetic substance and T = absolute temperature.

Hysteresis

The lagging of intensity of magnetisation (I) or magnetic induction (B) behind magnetising field (H), when a specimen of a magnetic substance is taken through a complete cycle of magnetisation is called hysteresis.

Retentivity or Residual Magnetism

The value of the intensity of magnetisation of a material, when the magnetising field is reduced to zero is called retentivity or residual magnetism of the material.

Coercivity

The value of the reverse magnetising field that should be applied to a given sample in order to reduce its intensity of magnetisation or magnetic induction to zero is called coercivity.

Permanent Magnets

Commonly steel is used to make a permanent magnet because steel has high residual magnetism and high coercivity.

Electromagnets

Electromagnets are made of soft iron because area of hysteresis loop for soft iron is small. Therefore, energy loss is small for a cycle of magnetisation and demagnetisation.

(Permanent magnets are made by the materials such as steel, for which residual magnetism as well as coercivity should be high. Electromagnets are made by the materials such as soft iron for which residual magnetism is high, coercivity is low and hysteresis loss is low).

Important Points

• Magnetic length = 5 / 6 * geometric length of magnet.
• About 90% of magnetic moment is due to spin motion of electrons and remaining 10% of magnetic moment is due to the orbital motion of electrons.
• When a magnet having magnetic moment M is cut into two equal parts

(i) Parallel to its length

M’ = m / 2 * l = M / 2

(ii) Perpendicular to its length

M’ = m * l / 2 = M / 2

• When a magnet of length I, pole strength m and of magnetic moment M is turned into a semicircular arc then it new magnetic moment

M’ = m * 2R = m * 2 * 1 / π (πR = I)

= 2M / π (M = m * l)

• A thin magnet of moment M is turned into an arc of 90°. Then new magnetic moment

M’ = 2√2M / π

• A thin magnet of moment M is turned at mid point 90°.Then new magnet moment

M’ = M / √2

• A thin magnet of moment M is turned into an arc of 60°. Then new magnetic moment

M = 3M / π

• A thin magnet of moment M is bent at mid point at angle 60°. Then new magnetic moment.

M’ = M / 2

• Original magnet MOS is bent at O, the mid point at 60°.All sidesare equal
• The mutual interaction force between two small magnets of moments M₁ and M₂ is given by

F = K 6M₁M₂ / d⁴ in end- on position.

Here,d denotes the separation between magnets.

• Magnetic length = 5 / 6 * geometric length of magnet.
• Cause of diamagnetism is orbital motion and cause of paramagnetism is spin motion of electrons. Cause of ferromagnetism lies in formation of domains.
• The perpendicular bisector of magnetic axis is known as neutral axis of magnet. Magnetism at neutral axis is zero and at poles is maximum.
• For steel coercivity is large. However, retentivity is comparatively smaller in case of steel. So, steel is used to make permanent magnets.
• For soft iron, coercivity is very small and area of hysteresis loop is small. So, soft iron is an ideal material for making electromagnets.