A magnetic field is produced in the surrounding of any current carrying conductor.

The direction of this magnetic field can be obtained by Ampere’s swimming rule.

SI unit of magnetic field is Wm^-2 or T (telsa).

The strength of magnetic field is called one tesla, if a charge of one coulomb, when moving with a velocity of 1 ms^-1 along a direction perpendicular to the direction of the magnetic field experiences a force of one newton.

1 tesla (T) = 1 weber metre^-2 (Wbm^-2)

= 1 newton ampere^-1 metre^-1 (NA^-1 m^-1)

CGS units of magnetic field are called gauss or oersted.

1 gauss = 10^-4 tesla.

Maxwell’s Cork Screw Rule

If a right handed cork screw is imagined to be rotated in such a direction that tip of the screw points in the direction of the current, then direction of rotation of thumb gives the direction of magnetic line of force.

The conventional sign for a magnetic field coming out of the plane and normal to it is a dot i.e.,

The magnetic field perpendicular to the plane in the downward action is denoted by ®.

Ampere’s Swimming Rule

If a man is swimming along the wire in the direction of current his face turned towards the needle, so that the current enters through his feet, then north pole of the magnetic needle will be deflected towards his left hand.

Magnetic Field

The space in the surrounding of a magnet or any current carrying conductor in which its magnetic influence can be experienced.

Biot Savart’s Law

The magnetic field produced by a current carrying element of length dl, carrying current I at a point separated by a distance r is given by

dB = μ_o / 4 π Idl * r / r³

or dB = μ_o / 4 π Idl sin θ / r²

where, θ is the angle between the direction of the current and μ_o is absolute permeability of the free space.

SI unit of magnetic field is Wm^-2 or (tesla) and CGS unit of magnetic field is gauss or oersted 1 gauss = 10^-4tesla.

The direction of magnetic field dB is that of I dl * r .

Magnetic Field Due to a Straight Current Carrying Conductor

where φ1 and φ2 are angles, which the lines joining the two ends of the conductor to the observation point make with the perpendicular from the observation point to the conductor.

For infinite length conductor and observation point is near the centre of the conductor,

B = μ_o / 4 π 2I / r

for infinite length conductor and observation point is near one end of the conductor,

B = μ_o / 4 π I / r

Magnetic Field Lines

• They are used to res present magnetic B field in a region.
• They are closed continuous curves.
• Tangent drawn at any point gives the direction of magnetic field.
• They cannot interact.
• Outside a magnet, they are directed from north to south pole and inside a magnet they are directed from south to north.

The magnetic field lines due to a straight current carrying conductor are concentric circles having centre at conductor and in a plane perpendicular to the conductor.

The direction of magnetic field lines can be obtained by Right Hand Thumb Rule

Right Hand Thumb Rule

If we hold a current carrying conductor in the grip of the right hand in such a way that thumb points in the direction of current, then curling of fingers represents the direction of magnetic field lines.

Magnetic Field on the Axis of a Current Carrying Circular Coil

Magnetic field at axis at a distance x from centre O.

where, r = radius of the coil, n = number of turns in the coil and I = current,

At centre of the coil, B = μ_o nl / 2 r

If we look at one face of the coil and the direction of current flowing through the coil is clockwise, then that face has south polarity and if direction of current is anti-clockwise, then that face has north polarity.

Magnetic Dipole

Every current carrying loop is a magnetic dipole. It has two poles south
(S) and north (N).

This is similar to a bar magnet.

Each magnetic dipole has some magnetic moment (M). The magnitude of M is,

|M| = NiA

where, N = number of turns in the loop,

i = current in the loop and

A = area of cross-section of the loop.

The current carrying loop behaves as a small magnetic dipole placed along the axis one face of the loop behaves as north pole while the other face of loop behaves as south pole.

Ampere’s Circuital Law

The line integral of magnetic field induction B around any closed path in vacuum is equal to 110 times the total current threading the closed path, i.e.,

where B is the magnetic field, dl is small element, μ_o is the absolute permeability of free space and I is the current.

Ampere’s circuital law holds good for a closed path of any size and shape around a current carrying conductor because the relation is independent of distance form conductor.

Solenoid

A solenoid is a closely wound helix of insulated copper wire.

Magnetic field at a point well inside a long solenoid is given by

B = μ_o nl

where, n = number of turns per unit length and

I = current flowing through the solenoid

Magnetic field at a point on one end of a long solenoid is given by

B = μ_o nl / 2

Toroid

A toroidal solenoid is an anchor ring around which is large number of turns of a copper wire are wrapped.

A toroid is an endless solenoid in the form of a ring.

Magnetic field inside the turns of toroid is given by

B = μ_o nl

Magnetic field inside a toroid is constant and is always tangential to the circular closed path.

Magnetic field at any point inside the empty space surrounded by the toroid and outside the toroid, is zero, because net current enclosed by these space is zero.

Magnetic Field Due to a Current Carrying Long Circular Cylinder

Outside the cylinder (r > R)

B = μ / 2 π l / r

Inside the cylinder when it is made of a thin metal sheet,

B = O

Inside the cylinder when current is uniformly distributed throughout the cross-section of the cylinder (r < R)

B = μ_o μ_r / 2 π Ir / R²

where, μ_o and μ_r are permeabilities of free space and material of the cylinder, I is current flowing through the cylinder and r is radius of the cylinder.

Force Acting on a Charge Particle Moving in a Uniform Magnetic Field

F = q(v * B)

or F = |F| = Bqv sin θ

where, B = magnetic field intensity,

q = charge on particle,

u = speed of the particle and

θ = angle between magnetic field and direction of motion.

This force is perpendicular to B as well as v.

Its direction can be obtained from Fleming’s left hand rule.

Magnetic force acting on a current carrying conductor in a uniform magnetic field is given by

F = I ( I * B)

Fleming’s left Hand Rule

If we stretch the thumb, the forefinger and the central finger if left hand in such a way that all three are perpendicular to each other, then if forefinger represents the direction of magnetic field, central finger represents the direction of current flowing through the conductor, then thumb will represent the direction of magnetic force.

Lorentz Force

The total force experienced by a charge moving inside the electric and magnetic fields is called Lorentz force. It is given by F = q(E * v * B)

Motion of a Charged Particle in a Uniform Magnetic Field

When charged particle enter normal to the magnetic field it follows a circular path.

The radius of the path, r = mv / Bq

∴ r ∝ mv

and r ∝ 1 / (q / m)

Time period, T = 2πm / Bq

When charged particle enter magnetic field at any angle except 90°, then it follows helical path.

The radius of the path, r = mv sin θ / Bq

The distance travelled by the charged particle in one time period due to component of velocity v cos θ, is called pitch of the path

Pitch = T * v cos θ = 2πmv cos θ / Bq

Cyclotron

Cyclotron is a device used to accelerate positively charged particles such as proton, deuteron etc.

Principle of Cyclotron

A positively charged particle can be accelerated through a modera. electric field by crossing it again and again by use of strong magnetic field.

Radius of circular path, r = mv / Bq

Cyclotron frequency v = Bq / 2πm

where m and q are mass and charge of the positive ion and B is strength of the magnetic field.

Maximum kinetic energy gained by the particle.

E_max = B²q²r²_o / 2m

where, r_o = maximum radius of circular path.

When a positive ion is accelerated by the cyclotron, it moves with greater and greater speed. As the speed of ion becomes comparable with that of light, the mass of the ion increases according to the relation.

Where, m = mass of the lOD.
m_o = maximum mass of the ion.
v = speed of Ion and
c = speed of light.

Limitations of the Cyclotron

(i) Cyclotron cannot accelerated uncharged particle like neutron.

(ii) The positively charged particles having large mass i.e., ions cannot move at limitless speed in a cyclotron.

Force between Two Infinitely Long Parallel Current Carrying Conductors

The force is attractive if current In both conductors is in same direction and repulsive if current 10 both conductors is in opposite direction.

(if the currents is both parallel wires arc equal and In same direction, then magnetic field at a point exactly half way between the wire is zero.)

Torque acting on a Current Carrying Coil Placed Inside a Uniform Magnetic Field

Torque acting on a current carrying coil placed inside a uniform magnetic field is given

τ = NBIA sin θ
Where, N = number of turns in the coil,

E = magnetic field intensity,

I = current 10 the coil and

A = area of cross-section of the coil,

θ = angle between magnetic field and normal to the plane of the coil.

Moving Coil Galvanometer

It is a device used for the detection and measurement of the currents.

In equilibrium, deflecting torque = restoring torque

where, k = restoring torque per unit twist,

N = number of turns in the coil,

B = magnetic field intensity,

A = area of cross-section of the coil and

θ = angle of twist.

Current Sensitivity

The deflection produced per unit current in galvanometer is called its current sensitivity.

Current sensitivity

I_s = θ / I = NBA / K

Voltage Sensitivity

The deflection produced per unit voltage applied across the ends of galvanometer is called its voltage sensitivity.

Voltage sensitivity

V_s = θ / V = NBA / KR

where R is the resistance of the galvanometer.

Therefore for a sensitive galvanometer

(i) N should be large
(ii) B should be large
(iii) A should be large
(iv) K should be small

Ammeter

An ammeter is a low resistance galvanometer used for measuring the current in a circuit.

It is always connected in series.

Conversion of a Galvanometer into an Ammeter

A galvanometer can be converted into an ammeter by connecting a low resistance into its parallel.

If G is the resistance of a galvanometer and it give full scale deflection for current, I_g then required low resistance S, connected in its parallel for converting it into an ammeter of range I is given by

The resistance of an ideal ammeter is zero.

Voltmeter

A voltmeter is a high resistance galvanometer used for measuring the potential difference between two points.

It is always connected in parallel.

The resistance of an ideal voltmeter is infinity.

Conversion of a Galvanometer into a Voltmeter

A galvanometer can be converted into a voltmeter by connecting a high resistance into its series.

If a galvanometer of resistance G show full scale deflection for current I_g then required high resistance R, connected in series for converting it into a voltmeter of range V is given by

When current 1 flows through a conductor of resistance R for a time t then heat generated in it is given by

where, V = potential difference applied across the end, of the conductor.

Electric Power

The electrical energy produced or consumed per unit time is called electric power.

or

The rate at which work is done by the source of emf in maintaining the electric current in a circuit is called electric power of the circuit.

Electric Power, P = VI = I²R

= (V²/R)

where, V is the potential difference across the conductor, I is flowing through the conductor and R is the resistance.

Its SI unit is watt (W).

The other units of electric power are kilowatt and horse power.

• 1 kW = 1000 W
• 1 HP = 746 W

Electric Energy

The energy supplied by any source in maintaining the current in the electric circuit is called electric energy consumed by the electric circuit.

Its SI unit is joule (J) but another unit is watt-hour. The bigger unit of electric energy is kilowatt hour (kWh). It is known as board of trade (SOT) unit.

1 kilowatt hour = 1000 watt x 1 hour
= 1000 J/s x 3600 s
= 3.6 x 10⁶ J
1 Horse power = 746 watt

The electric energy consumed in kWh is given by

• The electrical resistance of the wires supplying current is very small, therefore these wire do not heat up when current passes through them.
• The electrical resistance of filament of a lamp is very high, therefore it shows more heating effect, when electric current passes through it.
• A heater wire must be of high resistivity and of high melting point.

Electric Fuse

An electric fuse is a safety device used for protecting electric circuits bum demaging it due to excess flow of current. It is made of tin-lead alloy (63% tin + 37% lead).

It should have high resistance and low melting point and should be connected in series with the live wire.

Maximum safe current which can be passed through a fuse wire is it dependent of its length. However, it depends on the radius r of wire as

I ∝ r^(3/2)

Short Circuiting

When accidentily the live wire comes in contact with nutral wire, resistance of the circuit decreases and a high current flows through circuit. This phenomena is called short circuiting.

Overloading

When a high current flows through the wire which is beyond the ra of wire, then heating of wire takes place. This phenomena is ca overloading.

Rating of Electrical Appliances

The values of power and voltage taken together for an el appliance is called rating of the appliance.

• When a 40 W and a 100 W bulbs are connected in series, then 40 W bulb will glow brighter than 100 W bulb.
• When a 40 W and a 100 W bulbs are connected in parallel, then 100 W bulb will glow brighter than 40 W bulb.
• In series, if any bulb gets fused, then others will not glow.
• In parallel, if any one bulb get fused, then others will continue to glow.
• All switches should be connected in series with a live wire.

Fusing of Bulb When it is Switched On

Usually filament bulbs get fused when they are switched on. This is because with rise in temperature the resistance of the bulb increases and becomes constant in steady state. So, the power consumped by the bulb (V²R) initially is more than that in steady state and hence the bulb glows more brightly in beginning and may get fused.

Chemical Effect of Electric Current

When a direct current flows through a acidic or basic solution it dissociate into positive and negative ions. This phenomena is called electrolysis and these liquids are called electrolytes.

Some Terms of Electrolysis

1. Anode The electrode connected to the positive terminal battery is called an anode.
2. Cathode The electrode connected to the negative terminal of the battery is called a cathode.
3. Anions The ions carrying negative charge and move towards the anode in electrolysis are called anions.
4. Cations The ions carrying positive charge and move towards the cathode in electrolysis are called cations.
5. Voltameter The vessel in which the electrolysis is carried out is called a voltameter.

Faraday’s Laws of Electrolysis

First Law

The mass of the substance liberated at each electrode is directly proportional to the total charge passed through the electrolyte.

m ∝ q

Or M = Zq = ZIt

• where, Z = electrochemical equivalent of the substance deposited on electrode,
• q = total charge passed through the electrolyte,
• I = electric current and t = time.

Second Law

The mass of each substance liberated at the electrodes in electrolysis by a given amount of change is proportional to the chemical equivalents of the substances.

where m₁ and m₂ are masses of the substance liberated on electrodes by passing same amount of current for the same time and E₁, E₂ are chemical equivalents of these substances.

Faraday’s constant is equal to the amount of charge required to liberate one chemical equivalent (in gram) of mass of a substance at a electrode during electrolysis.

lta value is 96500 C/g equivalent.

Faraday’s constant, (F)= Ne

where N = Avogadro’s number and e = electronic charge.

Electroplating

The process of coating an object, that conduct electricity, with another metal is called electroplating. The article of cheep metal are coated with precious metal like silver and gold to make them look attractive,

Anodising

The process of coating aluminium with its oxide electrochemically to protect it against corrosion is called anodising.

Electrolysis is used for local anaesthesia. For it current is passed through a nerve, due to which it becomes insensitive to pain. Electrolysis is used for nerve stimulation of polio patients.

Electrochemical Cell

An electric cell is a device which maintains a continuous flow of charge in a circuit by a chemical reaction.

1. Primary Cell

Primary cell is an electrochemical cell, which once discharge cannot be put to use again by passing electric current from an external source.

(i) Voltaic This cell was first discovered in 1791 by scientist Galvani and Copper 1 i Zin:Rod 1800 by scientist Volta. To set up a simple cell, take a glass vessel containing dilute sulphuric cell, acid as the electrolyte. Two rods, one of copper and the other of zinc are kept immersed in the electrolyte. These rods are kept separated from each other. The” are called the electrodes. Due to chemical reaction a potential difference of 1.08 V is developed between these rod.

(ii) Dainel Cell It consists of a Zinc Rod copper vessel in which a porous pot containing dilute sulphuric acid is placed. An amalgamated zinc rod (which acts as the negative electrode) is put in the porous pot. The copper vessel contains a Dii. concentrated solution of copper sulphate which acts as an oxidising agent. To maintain the strength of copper sulphate solution, crystals of the copper sulphate are placed on the perforated shelf, partly immersed in the solution.

(iii) Leclanche Cell In this cell, anode is a carbon rod placed in a porous pot containing a mixture of powdered carbon and manganese dioxide. The carbon powder is used to make the manganese dioxide conducting or to increase the surface area of the carbon electrode. An amalgamated zinc rod is used as a negative electrode.

(iv) Dry Cell It is modified form of a Leclanche cell. The electrolyte in a dry cell is in the form of a jelly composed of starch, flour and ammonium chloride. The positive electrode is a carbon rod surrounded by a mixture of manganese dioxide and carbon contained in a cloth bag. This bag is placed in a zinc can and the space in between is filled with the ammonium chloride jelly. The zinc can itself acts as the negative electrode.

2. Secondary Cell

Secondary cell is an electrochemical cell, which has to be char initially by passing electric current from an external source, so as convert the stored chemical energy into electrical energy.

Lead accumulator, Ni-Fe or alkali accumulator are the secondary ce

Comparison of lead accumulator and Ni-Fe cell

Thermoelectric Effect

Thermocouple

The assembly of two different metals joined at their ends to have junctions in a circuit, is called a thermocouple.

Uses of Thermocouple

• Thermometer to measure temperature.
• Thermoelectric current-meter to measure current.
• Thermoelectric generator
• Thermoelectric refrigerator.

Seebeck Effect

When the two junctions of a thermocouple are kept at different temperatures, then an emf is produced between the junctions, which is called,thermo emf and this phenomena is called Seeback effect.

Thermoelectric Series or Seeback Series

Seeback form a series of metals for which thermoelectric current flows a thermocouple through the hot junction from a metal occuring earlier, to a metal occuring later, in series.

The series is given below

Bi, Ni, Co, Pd, Pt, Cu, Mn, Hg, Pb, Sn, Au, Zn, Cd, Fe, Sb, Te

Neutral Temperature (T_n)

The temperature of the hot junction at which thermo emf in a thermocouple is maximum, is called neutral temperature.

• The value of neutral temperature is a constant for thermocouple.
• Its value depends upon the nature of material forming the thermocouple.
• Its value is independent of the temperature of the cold junction.

Temperature of Inversion ( T_i)

The temperature of the hot junction at which the thermo emf in a thermocouple becomes zero and beyond it, reverses its direction is called temperature of inversion.

Its value depends upon the temperature of the cold junction as well as the nature of the materials forming the thermocouple.

Relation between neutral temperature (T„) and temperature of inversion ( Ti) is given by

Where T₀ is the temperature of cold junction.

When cold junction is at 0°C, then thermo emf changes with temperature of hot junction (T) at follows.

where α and β are constants depending upon nature of metals forming thermocouple.

Thermoelectric Power

The rate of change of thermo emf with temperature, thermoelectric power.

Thermoelectric Thermometer

It is a device used to measure both low and high temperatures. Thermoelectric thermometer have much wider range of measurement of temperature (from —200°C to 1600°C). They are quite sentitive and can measure temperature accurately upto 0.05°C. Disadvantage of thermometer is that it does not give direct reading and hence it cannot be used in experiments on calorimetry.

Peltier Effect

• When current is passed through a thermocouple, then heat is generated at its one junction and absorbed at another junction. This phenomena is called Peltier effect.
• Peltier effect is the reverse effect of Seeback effect.

Peltier Coefficient

The ratio of heat energy absorbed or evolved at a junction of a thermocouple to the charge flowing through it called Peltier coefficient.

Peltier coefficient,

Its SI unit is JC^-1‘.

Thomson’s Effect

If two parts of a single conductor are maintained at different temperatures, then an emf is produced between them, which is calico Thomson’s emf and this phenomena is called Thomson’s effect.

Thomson’s Coefficient

The ratio of Thomson’s emf and temperature difference between two points is called Thomson’s coefficient.

Thomson’s coefficient, (σ) = (dV/dT)

where, dV = potential difference between two points and dT = temperature difference between the two same points.

Thermopile

It is a combination of a large number of thermocouples in series. A’ such, it is able to detect the heat radiation and to note small variation or difference in temperature.

The rate of flow of charge through any cross-section of a wire is called electric current flowing through it.

Electric current (I) = q / t. Its SI unit is ampere (A).

The conventional direction of electric current is the direction of motion of positive charge.

The current is the same for all cross-sections of a conductor of non-uniform cross-section. Similar to the water flow, charge flows faster where the conductor is smaller in cross-section and slower where the conductor is larger in cross-section, so that charge rate remains unchanged.

If a charge q revolves in a circle with frequency f, the equivalent current,

i = qf

(In a metallic conductor current flows due to motion of free electrons while in electrolytes and ionized gases current flows due to electrons and positive ions.)

Types of Electric Current

According to its magnitude and direction electric current is of two types

(i) Direct Current (DC) Its magnitude and direction do not change with time. A ceil, battery or DC dynamo are the sources of direct current.

(ii) Alternating Current (AC) An electric current whose magnitude changes continuously and changes its direction periodically is called alternating current. AC dynamo is source of alternating current.

Current Density

The electric current flowing per unit area of cross-section of conductor is called current density.

Current density (J) = I / A

Its S1 unit is ampere metre^-2 and dimensional formula is [AT^-2].

It is a vector quantity and its direction is in the direction of motion positive charge or in the direction of flow of current.

Thermal Velocity of Free Electrons

Free electrons in a metal move randomly with a very high speed of the order of 10⁵ ms-1. This speed is called thermal velocity of free electron.

Average thermal velocity of free electrons in any direction remains zero.

Drift Velocity of Free Electrons

When a potential difference is applied across the ends of a conductor, the free electrons in it move with an average velocity opposite to direction of electric field. which is called drift velocity of free electrons.

Drift velocity v_d = eEτ / m = eVτ / ml

where, τ = relaxation time, e = charge on electron,

E = electric field intensity, 1 = length of the conductor,

V = potential difference across the ends of the conductor

m = mass of electron.

Relation between electric current and drift velocity is given by

v_d = I / An e

Mobility

The drift velocity of electron per unit electric field applied is mobility of electron.

Mobility of electron (μ) = v_d / E

Its SI unit is m²s^-1V^-1 and its dimensional formula is [M^-1T²A].

Ohm’s Law

If physical conditions of a conductor such as temperature remains unchanged, then the electric current (I) flowing through the conductor is directly proportional to the potential difference (V) applied across its ends.

I ∝ V

or V = IR

where R is the electrical resistance of the conductor and R = Ane² τ / ml.

Electrical Resistance

The obstruction offered by any conductor in the path of flow of current is called its electrical resistance.

Electrical resistance, R = V / I

Its SI unit is ohm (Ω) and its dimensional formula is [ML²T^-3A^-2].

Electrical resistance of a conductor R = ρl / A

where, l = length of the conductor, A = cross-section area and

ρ = resistivity of the material of the conductor.

Resistivity

Resistivity of a material of a conductor is given by

ρ = m / n² τ

where, n = number of free electrons per unit volume.

Resistivity of a material depend on temperature and nature of the material.

It is independent of dimensions of the conductor, i.e., length, area of cross-section etc.

Resistivity of metals increases with increase in temperature as

ρ_t = ρ_o (1 + αt)

where ρ_o and ρ_t are resistivity of metals at O°C and t°C and α temperature coefficient of resistivity of the material.

For metals α is positive, for some alloys like nichrome, manganin and constantan, α is positive but very low.

For semiconductors and insulators. α is negative.

Resistivity is low for metals, more for semiconductors and very high alloys like nichrome, constantan etc.

(In magnetic field the resistivity of metals increases. But resistivity of ferromagnetic materials such as iron, nickel, cobalt etc decreases in magnetic field.)

Electrical Conductivity

The reciprocal of resistivity is called electrical conductivity.

Electrical conductivity (σ) = 1 / ρ = 1 / RA = ne² τ / m

Its SI units is ohm^-1 m^-1 or mho m^-1 or siemen m^-1.

Relation between current density (J) and electrical conductivity (σ) is given by

J = σ E

where, E = electric field intensity.

Ohmic Conductors

Those conductors which obey Ohm’s law, are called ohmic conductors e.g., all metallic conductors are ohmic conductor.

For ohmic conductors V – I graph is a straight line.

Non-ohmic Conductors

Those conductors which do not obey Ohm’s law, are called non-ohmic conductors. e.g., diode valve, triode valve, transistor , vacuum tubes etc.

For non-ohmic conductors V – I graph is not a straight line.

Superconductors

When few metals are cooled, then below a certain critical temperature their electrical resistance suddenly becomes zero. In this state, these substances are called superconductors and this phenomena is called superconductivity.

Mercury become superconductor at 4.2 K, lead at 7.25 K and niobium at 9.2 K

Colour Coding of Carbon Resistors

The resistance of a carbon resistor can be calculated by the code given on it in the form of coloured strips.

This colour coding can be easily learned in the sequence “B B ROY Great Bratain Very Good Wife”.

Combination of Resistors

1.In Series

(i) Equivalent resistance, R = R₁ + R₂ + R₃

(ii) Current through each resistor is same.

(iii) Sum of potential differences across individual resistors is equal to the potential difference, applied by the source.

2. In Parallel

Equivalent resistance

1 / R = 1 /R₁ + 1 / R₂ + 1 / R₃

Potential difference across each resistor is same.

Sum of electric currents flowing through individual resistors is equal to the be electric current drawn from the source.

Electric Cell

An electric cell is a device which converts chemical energy into electrical energy.

Electric cells are of two types

(i) Primary Cells Primary ceUs cannot be charged again. Voltic, Daniel and Leclanche cells are primary cells.

(ii) Secondary Cells Secondary cells can be charged again and again. Acid and alkali accumulators are secondary cells.

Electro – motive – Force (emf) of a Cell

The energy given by a cell in flowing unit positive charge throughout the circuit completely one time, is equal to the emf of a cell.

Emf of a cell (E) = W / q.

Its SI unit is volt.

Terminal Potential Difference of a Cell

The energy given by a cell in flowing unit positive charge through till outer circuit one time from one terminal of the cell to the other terminal of the cell.

Terminal potential difference (V) = W / q.

Its SI unit is volt.

Internal Resistance of a Cell

The obstruction offered by the electrolyte of a cell in the path of electric current is called internal resistance (r) of the cell. Internal resistance of a cell

(i) Increases with increase in concentration of the electrolyte.

(ii) Increases with increase in distance between the electrodes.

(iii) Decreases with increase in area of electrodes dipped in electrolyte.

Relation between E. V and r

E = V + Ir

r = (E / V – 1) R

If cell is in charging state, then

E = V – Ir

Grouping of Cells

(i) In Series If n cells, each of emf E and internal resistance r are connected in series to a resistance R. then equivalent emf

E_eq = E₁ + E₂ + …. + E_n = nE

Equivalent internal resistance r_eq = r₁ + r₂ + …. + r_n = nr

Current In the circuit I = E_eq / (R + r_eq) = nE / (R + nr)

(ii) In Parallel If n cells. each of emf E and internal resistance r are connected to in parallel. then equivalent emf. E_eq = E

Equivalent internal resistance

1 / r_eq = 1 / r₁ + 1 / r₁ + … + 1 / r_n = n / r or r_eq = r / n

Current In the circuit I = E / (R + r / n)

(iii) Mixed Grouping of Cells If n cells, each of emf E and internal resistance r are connected in series and such m rows are connected in parallel, then

Equivalent emf, E_eq

Equivalent Internal resistance r_eq

Current in the circuit, I = nE / (R + nr / m)

or I = mnE / mR + nr

Note Current in this circuit will be maximum when external resistance is equal to the equivalent internal resistance, i.e.,

R = nr / m ⇒ mR = nr

Kirchhoff’s Laws

There are two Kirchhoff’s laws for solving complicated electrical circuits

(i) Junction Rule The algebraic sum of all currents meeting at a junction in a closed circuit is zero, i.e., Σ I = O.

This law follows law of conservation of charge.

(ii) Loop Rule The algebraic sum of all the potential differences in any closed circuit is zero, i.e.,

ΣV = 0 ⇒ ΣE = ΣIR

This law follows law of conservation of energy.

Balanced Wheatstone Bridge

Wheatstone bridge is also known as a metre bridge or slide wire bridge.

This is an arrangement of four resistance in which one resistance is unknown and rest known. The Wheatstone bridge as shown in figure. The bridge is said to be balanced when deflection in galvanometer is zero, i.e., i_g = O.

Principle of Wheatstone Bridge

P / Q = R / S

The value of unknown resistance S can found. as we know the value of P,Q and R. It may be remembered that the bridge is most sensitive, when all the four resistances are of the same order.

Meter Bridge

This is the simplest form of Wheatstone bridge and is specially useful for comparing resistance more accurately.

R / S = l₁ / (100 – l₁)

where l₁ is the length of wire from one end where null point is obtained.

Potentiometer

Potentiometer is an ideal device to measure the potential difference between two points. It consists of a long resistance wire AB of uniform cross section in which a steady direct current is set up by means of a battery.

If R be the total resistance of potentiometer wire L its total length, then potential gradient, i.e., fall in potential per unit length along the potentiometer will be

K = V / L = IR / L

= E_o R / (R_o + R)L

where, E_o = emf of battery and R_o = resistance inserted by means of rheostat Rh.

Determination of emf of a Cell using Potentiometer

If with a cell of emf E on sliding the contact point we obtain zero deflection in galvanometer G when contact point is at J at a length I from the end where positive terminal of cell have been joined. then fall in potential along length i is just balancing the emf of cell. Thus, we have

E = Kl

or E₁ / E₂ = l₁ / l₂

Determination of Internal Resistance of a Cell using Potentiometer

The arrangement is shown in figure. If the cell E is in open circuit and balancing length l₁, then

E = Kl₁

But if by inserting key K₂ circuit of cell is closed, then ooten difference V is balanced by a length l₂ of potential where

V = Kl₂

Internal resistance of cell

r = E – V / V , R = l₁ – l₂ / l₂ * R

Important Points

• Potentiometer is an ideal voltmeter.
• Sensitivity of potentiometer is increased by increasing length of potentiometer wire.
• If n identical resistances are first connected in series and then in parallel. the ratio of the equivalent resistance.

R_s / R_p= n² / 1

• If a skeleton cube is made with 12 equal resistance,each having a resistance R, then the net resistance across

1. The diagonal of cube = 5 / 6 R
2. The diagonal of a face = 3 / 4 R
3. along a side = 7 / 12 R

• If a resistance wire is stretched to a greater length, keeping volume constant, then

R ∝ l² ⇒ R₁ / R₂ = (l₁ / l₂)²

and R ∝ 1 / r⁴ ⇒ R₁ / R₂ = (r₂ / r₁)⁴

where l is the length of wire and r is the radius of cross-section area of wire.

Charge is that property of an object by virtue of which it apply electrostatic force of interaction on other objects.

Charges are of two types

(i) Positive charge

(ii) Negative charge

Like charges repel and unlike charges attract each other.

Quantization of Charge

Charge on any object can be an integer multiple of a smallest charge (e).

Q = ± ne

where, n = 1, 2, 3,……. and e = 1.6 * 10^-19 C.

Conservation of Charge

Charge can neither be created nor be destroyed. but can be transferral from one object to another object.

Recently a new particle has been discovered called ‘Quark’. It contains charge ± e / 3, ± 2e / 3.

[The protons and neutrons are combination of other entities called quarks, which have charges 1 / 3 e. However, isolated quarks have not been observed, so, quantum of charge is still e. ]

Coulomb’s Law of Electrostatics

Electrostatic force of interaction acting between two stationary charges is given by

F = 1 / 4π ε_o q₁q₂ / r²

where q₁, q₂ are magnitude of point charges, r is the distance between them and ε_o is permittivity of free space.

Here, 1 / 4πε_o = (10^-7 N – s² / C²)C²

Substituting value of c = 2.99792458 X 10⁸ m/s,

We get 1 / 4πε_o = 8.99 x 10⁹N-m²/C²

In examples and problems we will often use the approximate value,

1 / 4πε_o = 9 * 10⁹N-m²/C²

The value of ε_o is 8.85 * 10^-12 C² / N-mC².

If there is another medium between the point charges except air or vacuum, then ε_o is replaced by ε_oK or ε_oε_r or ε.

where K or ε_r is called dielectric constant or relative permittivity of the medium.

K = ε_r = ε / ε_o

where, ε = permittivity of the medium.

For air or vacuum, K = 1

For water, K = 81

For metals, K = ∞

Coulomb’s Law in Vector Form

Force on q₂ due to q₁,

The above equations give the Coulomb’s law in vector form.

Force on q₁ due to q₂ = – Force on q₂ due to q₁

F₁₂ = – F₂₁

F₁₂ = q₁q₂ / 4πε . r₁ – r₂ / |r₁ – r₂|³

The forces due to two point charges are parallel to the line joining point charges; such forces are called central forces and electrostatic forces are conservative forces.

Electric Field

The space in the surrounding of any charge in which its influence can be experienced by other charges is called electric field.

Electric Field Lines

“An electric field line is an imaginary line or curve drawn through a region of space so that its tangent at any point is in the direction of the electric field vector at that point. The relative closeness of the lines at some place give an idea about the intensity of electric field at that point.”

Two lines can never intersect.

Electric field lines always begin on a positive charge and end on a negative charge and do not start or stop in mid space.

Electric Field Intensity (E)

The electrostatic force acting per unit positive charge on a point in electric field is called electric field intensity at that point.

Electric field intensity E =

Its SI unit is NC^-1 or Vim and its dimension is [MLT^-3 A^-1].

It is a vector quantity and its direction is in the direction of electrostatic force acting on positive charge.

Electric field intensity due to a point charge q at a distance r is given by

E = 1 / 4π ε_o q / r²

Electric Potential (V)

Electric potential at any point is equal to the work done per positive charge in carrying it from infinity to that point in electric field.

Electric potential, V = W / q

Its SI unit is J / C or volt and its dimension is [ML²T^-3A^-1].

It is a scalar quantity.

Electric potential due to a point charge at a distance r is given by

v = 1 / 4π ε_o q / r

Potential Gradient

The rate of change of potential with distance in electric field is called potential gradient.

Potential gradient = dV / dr

Its unit is V / m.

Relation between potential gradient and electric field intensity is given by

E = – (dV / dr)

Equipotential Surface

Equipotential surface is an imaginary surface joining the points of same potential in an electric field. So, we can say that the potential difference between any two points on an equipotential surface is zero. The electric lines of force at each point of an equipotential surface are normal to the surface.

(i) Equipotential surface may be planer, solid etc. But equipotential surface can never be point size.

(ii) Electric field is always perpendicular to equipotential surface.

(iii) Equipotential surface due to an isolated point charge is spherical.

(iv) Equipotential surface are planer in an uniform electric field.

(v) Equipotential surface due to a line charge is cylindrical.

Electric Lines of Force

Electric lines of force are the imaginary lines drawn in electric field at Which a positive test charge will move if it is free to do so.

Electric lines of force start from positive charge and terminate on negative charge.

A tangent drawn at any point on electric field represents the direction of electric field at that point.

Two electric lines of force never intersect each other.

Electric lines of force are always perpendicular to an equipotential surface.

Electric Flux (φ_E)

Electric flux over an area is equal to the total number of electric field lines crossing this area.

Electric flux through a small area element dS is given by

φ_E = E. dS

where E= electric field intensity and dS = area vector.

Its SI unit is N – m²C^-1.

Gauss’s Theorem

The electric flux over any closed surface is 1 / ε_o times the total charge enclosed by that surface, i.e.,

If a charge q is placed at the centre of a cube, then

total electric flux linked with the whole cube = q / ε_o

electric flux linked with one face of the cube = q / 6 ε_o

(i) Electric Field at Any Point on the Axis of a Uniformly Charged Ring A ring-shaped conductor with radius a carries total charge Q uniformly distributed around it. Let us calculate the electric field at a point P that lies on the axis of the ring at distance x from its centre.

E_x = 1 / 4π ε_o * xQ / (x² + a²)^3/2

The maximum value of electric field

E_x = 1 / 4π ε_o (2Q / 3√3R²

(ii) Electric Field due to a Charged Spherical Shell

(a) At an extreme point (r > R)

E = 1 / 4π ε_o q / r²

(b) At the surface of a shell (r = R)

E = 1 / 4π ε_o q / R²

(c) At an internal point (r < R)

E = 0

(iii) Electric Potential due to a Charged Conducting Spherical Shell

(a) At an extreme point (r > R)

V = 1 / 4π ε_o q / r

(b) At the surface of a shell (r = R)

V = 1 / 4π ε_o q / R

(c) At an internal point (r < R)

V = 1 / 4π ε_o q / R

Therefore potential inside a charged conducting spherical shell equal to the potential at its surface.

(iv) Electric Field and Potential due to a Charged Non-Conducting Sphere

At an extreme point, (r > R)

(v) Electric Field Intensity due to an Infinite Line Charge

E = 1 / 2 π ε_o λ / r

where λ is linear charge density and r is distance from the line charge.

(vi) Electric Field Near an Infinite Plane Sheet of Charge

E = σ / 2 ε_o

where σ = surface charge density.

If infinite plane sheet has uniform thickness, then

E = σ / ε_o

Electric Dipole

An electric dipole consists of two equal and opposite point charges separated by a very small distance. e.g., a molecule of HCL, a molecule of water etc.

Electric Dipole Moment p = q * 2 l

Its SI unit is ‘coulomb-metre’ and its dimension is [LTA).

It is a vector quantity and its direction is from negative charge towards positive charge.

Electric Field Intensity and Potential due to an Electric Dipole

(i) On Axial Line

Electric field intensity E = 1 / 4 π ε_o * 2 pr / (r² – a²)²

If r > > 2a, then E = 1 / 4 π ε_o * 2 p / r³

Electric potential V = 1 / 4 π ε_o * p / (r² – a²)

Ifr > > 2a, then V = 1 / 4 π ε_o * p / r²

(ii) On Equatorial Line

Electric field intensity E = 1 / 4 π ε_o * p / (r² + a²)^{3 / 2}

If r > > 2a, then E = 1 / 4 π ε_o * p / r³

Electric potential V = 0

(iii) At any Point along a Line Making θ Angle with Axis

Electric field intensity E = 1 / 4 π ε_o * p √1 + 3 cos² θ / r³

Electric potential V = 1 / 4 π ε_o * p cos θ / (r² – a² cos² θ)

If r > > 2a, then V = 1 / 4 π ε_o * p cos θ / r²

Torque

Torque acting on an electric dipole placed in uniform electric field is given by

τ = Ep sin θ or τ = p * E

When θ = 90°, then ‘τ_max = Ep

When electric dipole is parallel to electric field, it is in stable equilibrium and when it is anti-parallel to electric field, it is in unstable equilibrium.

Work Done

Work done is rotating an electric dipole in a uniform electric field from angle θ₁ to θ₂ is given by

W = Ep (cos θ₁ – cos θ₂)

If initially it is in the direction of electric field, then work done in rotating through an angle θ, W = Ep (1 – cos θ).

Potential Energy

Potential energy of an electric dipole in a uniform electric field is given by U = – pE cos &theta.

Dipole in Non-uniform Electric Field

When an electric dipole is placed in a non-uniform electric field, then a resultant force as well as a torque act on it.

Net force on electric dipole = (qE₁ – qE₂), along the direction of greater e ecmc field intensity.

Therefore electric dipole undergo rotational as well as linear motion.

Potential Energy of Charge System

Two point charge system, contains charges q₁ and q₂ separated by a distance r is given by U = 1 / 4 π ε_o * q₁q₂ / r

Three point charge system

U = 1 / 4 π ε_o * [q₁q₂ / r₁ + q₂q₃ / r₂ + q₃q₂ / r₃

Important Points

When charge is given to a soap bubble its size gets increased.

In equilibrium for a charged soap bubble, pressure due to surface tension

= electric pressure due to charging

4T / r = σ² / 2 ε_o

or 4T / r = 1 / 2 ε_o (q / 4 πr²)²

or q = 8 πr √2 ε_o rT

where, r is radius of soap bubble and T is surface tension of soap bubble.

Behaviour of a Conductor in an Electrostatic Field

(i) Electric field at any point inside the conductor is zero.

(ii) Electric field at any point on the surface of charged conductor is directly proportional to the surface density of charge at that point, but electric potential does not depend upon the surface density of charge.

(iii) Electric potential at any point inside the conductor is constant and equal to potential.

Electrostatic Shielding

The process of protecting certain field from external electric field is called, electrostatic shielding.

Electrostatic shielding is achieved by enclosing that region in a closed metallic chamber.

Dielectric

Dielectrics are of two types Non-polar Dielectric The non-polar dielectrics (like N₂, O₂, benzene, methane) etc. are made up of non-polar atoms/molecules, in which the centre of positive charge coincides with the centre of negative charge of the atom/molecule.

Polar Dielectric

The polar dielectric (like H₂O, CO₂, NH₃ etc) are made up of polar atoms/molecules, in which the centre of positive charge does not coincide with the centre of negative charge of the atom.

Capacitor

A capacitor is a device which is used to store huge charge over it, without changing its dimensions.

When an earthed conductor is placed near a charged conductor, then it decreases its potential and therefore more charge can be stored over it.

A capacitor is a pair of two conductors of any shape, close to each other and have equal and opposite charges.

Capacitance of a conductor C = q / V

Its 81 unit is coulomb/volt or farad.

Its other units are 1 μ F = 10^-6 F

1 μμ F = 1 pF = 10^-12 F

Its dimensional formula is [M^-1L^-2T⁴A²].

Capacitance of an Isolated Spherical Conductor

C = 4 π ε_o K R

For air K = 1

∴ C = 4 π ε_o R = R / 9 * 10⁹

Parallel Plate Capacitor

The parallel plate capacitor consists of two metal plates parallel to each other and separated by a distance d.

Capacitance C = KA ε_o / d

For air capacitor C_o = A ε_o / d

When a dielectric slab is inserted between the plates partially, then its capacitance.

C = A ε_o / (d – t + t / K)

If a conducting (metal) slab is inserted between the plates, then

C = A ε_o / (d – t)

When more than one dielectric slabs are placed fully between the plates, then

The plates of a parallel plate capacitor attract each other with a force

F = Q² / 2 A ε_o

When 9. dielectric slab is placed between the plates of a capacitor than charge induced on its side due to polarization of dielectric is

q’ = q (1 – 1 / k)

Capacitors Combination

(i) In Series

Resultant capacitance = 1 / C = 1 / C₁ + 1 / C₂ + 1 / C₃ + ….

In series charge is same on each capacitor, which is equal to the charge supplied by the source.

If V₁, V₂, V₃,…. are potential differences across the plates of the capacitors then total voltage applied by the
source

V = V₁ + V₂ + V₃ + ….

(ii) In Parallel

Resultant capacitance C = C₁ + C₂ + C₃ + ….

In parallel potential differences across the plates of each capacitor is same.

If q₁, q₂, q₃ are charges on the plate of capacitors connected in parallel then total charge given by the source

q = q₁ + q₂ + q₃ + ….

Electric potential energy of a charged conductor or a capacitor is given by,

U = 1 / 2 Vq = 1 / 2 CV² = 1 / 2 q² / C

Redistribution of Charge

When two isolated charged conductors are connected to each other then charge is redistributed in the ratio of their capacitances.

Common potential V = q₁ + q₂ / C₁ + C₂ = C₁V₁ + C₂V₂ / C₁ + C₂

Energy loss = 1 / 2 C₁C₂ (V₁ – V₂)² / (C₁ + C₂)

This energy is lost in the form of heat in connecting wires.

When n, small drops, each of capacitance C, charged to potential V with charge q, surface charge density σ and potential energy U coalesce to from a single drop.

Then for new drop

Total charge = nq

Total capacitance = n^l/3C

Total potential = n^2/3 V

Surface charge density = n^l/3 σ

Total potential energy = n^2/3 U

Van-de-Graaff Generator

It is a device used to build up very high potential difference of the order of few million volt.

Its working is based on two points

(i) The action of sharp points (corona discharge)
(ii) Total charge given to a spherical shell resides on its outer surface.

Lightning Conductor

When a charged cloud passes by a tall building, the charge on cloud passes to the earth through the building. This causes a damage to the building. Thus to protect the tall building lightning, the lightning conductors, (which are pointed metal roe passes over the charge on the clouds to earth, thus protecting building.

A force which produces a change in configuration of the object on applying it, is called a deforming force.

Elasticity

Elasticity is that property of the object by virtue of which it regain its original configuration after the removal of the deforming force.

Elastic Limit

Elastic limit is the upper limit of deforming force upto which, if deforming force is removed, the body regains its original form completely and beyond which if deforming force is increased the body loses its property of elasticity and get permanently deformed.

Perfectly Elastic Bodies

Those bodies which regain its original configuration immediately and completely after the removal of deforming force are called perfectly elastic bodies. e.g., quartz and phosphor bronze etc.

Perfectly Plastic Bodies

Those bodies which does not regain its original configuration at all on the removal of deforming force are called perfectly plastic bodies, e.g., putty, paraffin, wax etc.

Stress

The internal restoring force acting per unit area of a deformed body is called stress.

Stress = Restoring force / Area

Its unit is N/m² or Pascal and dimensional formula is [ML^-12T^-2].

Stress is a tensor quantity.

Stress is of Two Types

(i) Normal Stress If deforming force is applied normal to the area, then the stress is called normal stress.

If there is an increase in length, then stress is called tensile stress.

If there is a decrease in length, then stress is called compression stress.

(ii) Tangential Stress If deforming force is applied tangentially, then the stress is called tangential stress.

Strain

The fractional change in configuration is called strain.

Strain = Change in the configuration / Original configuration

It has no unit and it is a dimensionless quantity.

According to the change in configuration, the strain is of three types

(1) Longitudinal strain= Change in length / Original length

(2) Volumetric strain = Change in volume / Original volume

(iii) Shearing strain = Angular displacement of the plane perpendicular to the fixed surface.

Hooke’s Law

Within the limit of elasticity, the stress is proportional to the strain.

Stress &infi; Strain

or Stress = E * Strain

where, E is the modulus of elasticity of the material of the body.

Types of Modulus of Elasticity

1. Young’s Modulus of Elasticity

It is defined as the ratio of normal stress to the longitudinal strain Within the elastic limit.

y = Normal stress / Longitudinal strain

y = FΔl / Al = Mg Δl / πr²l

Its unit is N/m² or Pascal and its dimensional formula is [ML^-1T^-2].

2. Bulk Modulus of Elasticity

It is defined as the ratio of normal stress to the volumetric strain within the elastic limit.

K = Normal stress / Volumetric strain

K = FV / A ΔV = &DElta;p V / Δ V

where, Δp = F / A = Change in pressure.

Its unit is N/m² or Pascal and its dimensional formula is [ML^-1T^-2].

3. Modulus of Rigidity (η)

It is defined as the ratio of tangential stress to the shearing strain, within the elastic limit.

η = Tangential stress / Shearing strain

Its unit is N/m² or Pascal and its dimensional formula is [ML^-1T^-2].

Compressibility

Compressibility of a material is the reciprocal of its bulk modulus of elasticity.

Compressibility (C) = 1 / k

Its SI unit is N^-1m² and CGS unit is dyne^-1 cm².

Steel is more elastic than rubber. Solids are more elastic and gases are least elastic.

For liquids. modulus of rigidity is zero.
Young’s modulus (Y) and modulus of rigidity (η) are possessed by solid materials only.

Limit of Elasticity

The maximum value of deforming force for which elasticity is present in the body is called its limit of elasticity.

Breaking Stress

The minimum value of stress required to break a wire, is called breaking stress.

Breaking stress is fixed for a material but breaking force varies with area of cross-section of the wire.

Safety factor = Breaking stress / Working stress

Elastic Relaxation Time

The time delay in restoring the original configuration after removal of deforming force is called elastic relaxation time.

For quartz and phosphor bronze this time is negligible.

Elastic After Effect

The temporary delay in regaining the original configuration by the elastic body after the removal of deforming force, is called elastic after effect.

Elastic Fatigue

The property of an elastic body by virtue of which its behaviour becomes less elastic under the action of repeated alternating deforming force is called elastic fatigue.

Ductile Materials

The materials which show large plastic range beyond elastic limit are called ductile materials, e.g., copper, silver, iron, aluminum, etc.

Ductile materials are used for making springs and sheets.

Brittle Materials

The materials which show very small plastic range beyond elastic limit are called brittle materials, e.g., glass, cast iron, etc.

Elastomers

The materials for which strain produced is much larger than the stress applied, with in the limit of elasticity are called elastomers, e.g., rubber, the elastic tissue of aorta, the large vessel carrying blood from heart. etc.

Elastomers have no plastic range.

Elastic Potential Energy in a Stretched Wire

The work done in stretching a wire is stored in form of potential energy of the wire.

Potential energy U = Average force * Increase in length

= 1 / 2 FΔl

= 1 / 2 Stress * Strain * Volume of the wire

Elastic potential energy per unit volume

U = 1 / 2 * Stress * Strain

= 1 / 2 (Young’s modulus) * (Strain)²

Elastic potential energy of a stretched spring = 1 / 2 kx²

where, k = Force constant of spring and x = Change in length.

Thermal Stress

When temperature of a rod fixed at its both ends is changed, then the produced stress is called thermal stress.

Thermal stress = F / A = yαΔθ

where, α = coefficient of linear expansion of the material of the rod.

When temperature of a gas enclosed in a vessel is changed, then the thermal stress produced is equal to change in pressure (Δp)of the gas.

Thermal stress = Δ p = Ky Δ θ

where, K = bulk modulus of elasticity and

γ = coefficient of cubical expansion of the gas.

Interatomic force constant

K = Yr_o

where, r_o = interatomic distance.

Poisson’s Ratio

When a deforming force is applied at the free end of a suspended wire of length 1 and radius R, then its length increases by dl but its radius decreases by dR. Now two types of strains are produced by a single force.

(i) Longitudinal strain = &DElta;U l

(ii) Lateral strain = – Δ R/ R

∴ Poisson’s Ratio (σ) = Lateral strain / Longitudinal strain = – Δ R/ R / ΔU l

The theoretical value of Poisson’s ratio lies between – 1 and 0.5. Its practical value lies between 0 and 0.5.

Relation Between Y, K, η and σ

(i) Y = 3K (1 – 2σ)

(ii) Y = 2 η ( 1 + σ)

(iii) σ = 3K – 2η / 2η + 6K

(iv) 9 / Y = 1 / K + 3 / η or Y = 9K η / η + 3K

Important Points

• Coefficient of elasticity depends upon the material, its temperature and purity but not on stress or strain.
• For the same material, the three coefficients of elasticity γ, η and K have different magnitudes.
• Isothermal elasticity of a gas E_T = ρ where, ρ = pressure of the gas.
• Adiabatic elasticity of a gas E_s = γρ

where, γ = C_p / C_v ratio of specific heats at constant pressure and at constant volume.

• Ratio between isothermal elasticity and adiabatic elasticity E_s/ E_T = γ = C_p / C_v

Cantilever

A beam clamped at one end and loaded at free end is called a cantilever.

Depression at the free end of a cantilever is given by

δ = wl³ / 3YI_G

where, w = load, 1 = length of the cantilever,

y = Young’s modulus of elasticity, and I_G = geometrical moment of inertia.

For a beam of rectangular cross-section having breadth b and thickness d.

I_G = bd³ / 12

For a beam of circular cross-section area having radius r,

I_G = π r⁴ / 4

Beam Supported at Two Ends and Loaded at the Middle

Depression at middle δ = wl³ / 48YI_G

Torsion of a Cylinder

where, η = modulus of rigidity of the material of cylinder,

r = radius of cylinder,

and 1 = length of cylinder,

Work done in twisting the cylinder through an angle θ

W = 1 / 2 Cθ²

Relation between angle of twist (θ) and angle of shear (φ)

rθ = lφ or φ = r / l = θ

Gravitation is one of the four classes of interactions found in nature.

These are

(i) the gravitational force

(ii) the electromagnetic force

(iii) the strong nuclear force (also called the hadronic force).

(iv) the weak nuclear forces.

Although, of negligible importance in the interactions of elementary particles, gravity is of primary importance in the interactions of objects. It is gravity that holds the universe together.

Newton’s Law of Gravitation

Gravitational force is a attractive force between two masses m₁ and m₂ separated by a distance r.

The gravitational force acting between two point objects is proportional to the product of their masses and inversely proportional to the square of the distance between them.

Gravitational force.

where G is universal gravitational constant.

The value of G is 6.67 X 10^-11 Nm² kg^-2 and is same throughout the universe.

The value of G is independent of the nature and size of the bodies well as the nature of the medium between them.

Dimensional formula of Gis [M^-1L³T^-2].

Important Points about Gravitation Force

(i) Gravitational force is a central as well as conservative force.

(ii) It is the weakest force in nature.

(iii) It is 1036 times smaller than electrostatic force and 10’l8times smaller than nuclear force.

(iv) The law of gravitational is applicable for all bodies, irrespective of their size, shape and position.

(v) Gravitational force acting between sun and planet provide it centripetal force for orbital motion.

(vi) Gravitational pull of the earth is called gravity.

(vii) Newton’s third law of motion holds good for the force of gravitation. It means the gravitation forces between two bodies are action-reaction pairs.

Following three points are important regarding the gravitational force

(i) Unlike the electrostatic force, it is independent of the medium between the particles.

(ii) It is conservative in nature.

(iii) It expresses the force between two point masses (of negligible volume). However, for external points of spherical bodies the whole mass can be assumed to be concentrated at its centre of mass.

Note Newton’s law of gravitation holde goods for object lying at uery large distances and also at very short distances. It fails when the distance between the objects is less than 10-9 m i.e., of the order of intermolecular distances.

Acceleration Due to Gravity

The uniform acceleration produced in a freely falling object due to the gravitational pull of the earth is known as acceleration due to gravity.

It is denoted by g and its unit is m/s². It is a vector quantity and its direction is towards the centre of the earth.

The value of g is independent of the mass of the object which is falling freely under gravity.

The value of g changes slightly from place to place. The value of g is taken to be 9.8 m/s² for all practical purposes.

The value of acceleration due to gravity on the moon is about. one sixth of that On the earth and on the sun is about 27 times of that on the earth.

Among the planets, the acceleration due to gravity is minimum on the mercury.

Relation between g and a is given by

g = Gm / R²

where M = mass of the earth = 6.0 * 10²⁴ kg and R = radius of the earth = 6.38 * 10⁶ m.

Acceleration due to gravity at a height h above the surface of the earth is given by

g_h = Gm / (R+h)² = g (1 – 2h / R)

Factors Affecting Acceleration Due to Gravity

(i) Shape of Earth Acceleration due to gravity g &infi; 1 / R² Earth is elliptical in shape. Its diameter at poles is approximately 42 km less than its diameter at equator.

Therefore, g is minimum at equator and maximum at poles.

(ii) Rotation of Earth about Its Own Axis If ω is the angular velocity of rotation of earth about its own axis, then acceleration due to gravity at a place having latitude λ is given by

g’ = g – Rω² cos² λ

At poles λ = 90° and g’ = g

Therefore, there is no effect of rotation of earth about its own axis at poles.

At equator λ = 0° and g’ = g – Rω²

The value of g is minimum at equator

If earth stapes its rotation about its own axis, then g will remain unchanged at poles but increases by Rω² at equator.

(iii) Effect of Altitude The value of g at height h from earth’s surface

g’ = g / (1 + h / R)²

Therefore g decreases with altitude.

(iv) Effect of Depth The value of gat depth h A from earth’s surface

g’ = g * (1 – h / R)

Therefore g decreases with depth from earth’s surface.

The value of g becomes zero at earth’s centre.

Gravitational Field

The space in the surrounding of any body in which its gravitational pull can be experienced by other bodies is called gravitational field.

Intensity of Gravitational Field

The gravitational force acting per unit mass at Earth any point in gravitational field is called intensity of gravitational field at that point.
It is denoted by E_g or I.

E_g or I = F / m

Intensity of gravitational field at a distance r from a body of mass M is given by

E_g or I = GM / r²

It is a vector quantity and its direction is towards the centre of gravity of the body.

Its S1 unit is N/m and its dimensional formula is [LT^-2].

Gravitational mass M_g is defined by Newton’s law of gravitation.

M_g = F_g / g = W / g = Weight of body / Acceleration due to gravity

∴ (M₁)g / (M₂)g = F_g1g2 / F_g2g1

Gravitational Potential

Gravitational potential at any point in gravitational field is equal the work done per unit mass in bringing a very light body from infinity to that point.

It is denoted by V_g.

Gravitational potential, V_g = W / m = – GM / r

Its SI unit is J / kg and it is a scalar quantity. Its dimensional formula is [L³r^-2].

Since work W is obtained, that is, it is negative, the gravitational potential is always negative.

Gravitational Potential Energy

Gravitational potential energy of any object at any point in gravitational field is equal to the work done in bringing it from infinity to that point. It is denoted by U.

Gravitational potential energy U = – GMm / r

The negative sign shows that the gravitational potential energy decreases with increase in distance.

Gravitational potential energy at height h from surface of earth

U_h = – GMm / R + h = mgR / 1 + h/R

Satellite

A heavenly object which revolves around a planet is called a satellite. Natural satellites are those heavenly objects which are not man made and revolve around the earth. Artificial satellites are those neaven objects which are man made and launched for some purposes revolve around the earth.

Time period of satellite

T = 2π √r³ / GM

= 2π √(R + h)³ / g [ g = GM / R²

Near the earth surface, time period of the satellite

T = 2π √R³ / GM = √3π / Gp

T = 2π √R / g = 5.08 * 10³ s = 84 min.

where p is the average density of earth.

Artificial satellites are of two types :

1. Geostationary or Parking Satellites

A satellite which appears to be at a fixed position at a definite height to an observer on earth is called geostationary or parking satellite.

Height from earth’s surface = 36000 km

Radius of orbit = 42400 km

Time period = 24 h

Orbital velocity = 3.1 km/s

Angular velocity = 2π / 24 = π / 12 rad / h

There satellites revolve around the earth in equatorial orbits.

The angular velocity of the satellite is same in magnitude and direction as that of angular velocity of the earth about its own axis.

These satellites are used in communication purpose.

INSAT 2B and INSAT 2C are geostationary satellites of India.

2. Polar Satellites

These are those satellites which revolve in polar orbits around earth. A polar orbit is that orbit whose angle of inclination with equatorial plane of earth is 90°.

Height from earth’s surface = 880 km

Time period = 84 min

Orbital velocity = 8 km / s

Angular velocity = 2π / 84 = π / 42 rad / min.

There satellites revolve around the earth in polar orbits.

These satellites are used in forecasting weather, studying the upper region of the atmosphere, in mapping, etc.

PSLV series satellites are polar satellites of India.

Orbital Velocity

Orbital velocity of a satellite is the minimum velocity required to the satellite into a given orbit around earth.

Orbital velocity of a satellite is given by

v_o = √GM / r = R √g / R + h

where, M = mass of the planet, R = radius of the planet and h = height of the satellite from planet’s surface.

If satellite is revolving near the earth’s surface, then r = (R + h) =- R

Now orbital velocity,

v_o = √gR

= 7.92km / h

if v is the speed of a satellite in its orbit and v_o is the required orbital velocity to move in the orbit, then

(i) If v < v_o, then satellite will move on a parabolic path and satellite falls back to earth.

(ii) If V = v_o then satellite revolves in circular path/orbit around earth.

(iii) If v_o < V < v_e then satellite shall revolve around earth in elliptical orbit.

Energy of a Satellite in Orbit

Total energy of a satellite

E = KE + PE

= GMm / 2r + (- GMm / r)

= – GMm / 2r

Binding Energy

The energy required to remove a satellite from its orbit around the earth (planet) to infinity is called binding energy of the satellite.

Binding energy of the satellite of mass m is given by

BE = + GMm / 2r

Escape Velocity

Escape velocity on earth is the minimum velocity with which a body has to be projected vertically upwards from the earth’s surface so that it just crosses the earth’s gravitational field and never returns.

Escape velocity of any object

v_e = √2GM / R

= √2gR = √8πp GR² / 3

Escape velocity does not depend upon the mass or shape or size of the body as well as the direction of projection of the body.

Escape velocity at earth is 11.2 km / s.

Some Important Escape Velocities

Relation between escape velocity and orbital velocity of the satellite

v_e = √2 v_o

If velocity of projection U is equal the escape velocity (v = v_e), then the satellite will escape away following a parabolic path.

If velocity of projection u of satellite is greater than the escape velocity ( v > v_e), then the satellite will escape away following a hyperbolic path.

Weightlessness

It is a situation in which the effective weight of the body becomes zero,

Weightlessness is achieved

(i) during freely falling under gravity

(ii) inside a space craft or satellite

(iii) at the centre of the earth

(iv) when a body is lying in a freely falling lift.

Kepler’s Laws of Planetary Motion

(i) Law of orbit Every planet revolve around the sun in elliptical orbit and sun is at its one focus.

(ii) Law of area The radius vector drawn from the sun to a planet sweeps out equal areas in equal intervals of time, i.e., the areal velocity of the planet around the sun is constant.

Areal velocity of a planet

dA / dt = L / 2m = constant

where L = angular momentum and m = mass of the planet.

(iii) Law of period The square of the time period of revolution of planet around the sun is directly proportional to the cube semi-major axis of its elliptical orbit.

T² &infi; a³ or (T₁ / T₂)² = (a₁ / a₂)³

where, a = semi-major axis of the elliptical orbit.

Important Points

(i) A missile is launched with a velocity less than the escape velocity. The sum of its kinetic energy and potential energy is negative.

(ii) The orbital speed of jupiter is less than the orbital speed of earth.

(iii) A bomb explodes on the moon. You cannot hear the sound of the explosion on earth.

(iv) A bottle filled with water at 30°C and fitted with a cork is taken to the moon. If the cork is opened at the surface of the moon then water will boil.

(v) For a satellite orbiting near earth’s surface

(a) Orbital velocity = 8 km / s

(b) Time period = 84 min approximately

(c) Angular speed ω = 2π / 84 rad / min

= 0.00125 rad / s

(vi) Inertial mass and gravitational mass

(a) Inertial mass = force / acceleration

(b) Gravitational mass = weight of body / acceleration due to gravity

(c) They are equal to each other in magnitude.

(d) Gravitational mass of a body is affected by the presence of other bodies near it. Inertial mass of a body remains unaffected by the presence of other bodies near it.

A wave is a vibratory disturbance in a medium which carries energy from one point to another point without any actual movement of the medium. There are three types of waves

1. Mechanical Waves Those waves which require a material medium for their propagation, are called mechanical waves, e.g., sound waves, water waves etc.
2. Electromagnetic Waves Those waves which do not require a material medium for their propagation, are called electromagnetic waves, e.g., light waves, radio waves etc.
3. Matter Waves These waves are commonly used in modern technology but they are unfamiliar to us. Thses waves are associated with electrons, protons and other fundamental particles.

Nature of Waves

(i) Transverse waves A wave in which the particles of the medium vibrate at right angles to the direction of propagation of wave, is called a transverse wave.

These waves travel in the form of crests and troughs.

(ii) Longitudinal waves A wave in which the particles of the medium vibrate in the same direction in which wave is propagating, is called a longitudinal wave.

These waves travel in the form of compressions and rarefactions.

Some Important Terms of Wave Motion

1. (i) Wavelength The distance between two nearest points in a wave which are in the same phase of vibration is called the wavelength (λ).
2. (ii) Time Period Time taken to complete one vibration is called time period (T).
3. (iii) Frequency The number of vibrations completed in one second is called frequency of the wave.
   
   Its SI unit is hertz.
4. (iv) Velocity of Wave or Wave Velocity The distance travelled by a wave in one second is called velocity of the wave (u).
   Relation among velocity, frequency and wavelength of a wave is given by v = fλ.
5. (v) Particle Velocity The velocity of the particles executing SHM is called particle velocity.

Sound Waves

Sound waves of all the mechanical waves that occur in nature, the most important in our everyday lives are longitudinal waves in a medium, usually air, called sound waves.

Sound waves are of three types

(i) Infrasonic Waves The sound waves of frequency lies between 0 to 20 Hz are called infrasonic waves.

(ii) Audiable Waves The sound waves of frequency lies between 20 Hz to 20000 Hz are called audiable waves.

(iii) Ultrasonic Waves The sound waves of frequency greater than 20000 Hz are called ultrasonic waves.

Sound waves are mechanical longitudinal waves and require medium for their propagation. Sound waves can travel through

[sound waves cannot propagate through vacuum.

If V_s, V_i and V_g are speed of sound waves in solid, liquid and gases, then

V_s > V_i > V_g

Sound waves (longitudinal waves) can reflect, refract, interfere and diffract but cannot be polarised as only transverse waves can polarised.]

Velocity of Longitudinal (Sound) Waves

Velocity of longitudinal (sound) wave in any medium is given by

where, E is coefficient of elasticity of the medium and ρ is density of the medium.

Newton’s Formula

According to Newton, the propagation of longitudinal waves in a gas is an isoth. ermal process. Therefore, velocity of longitudinal (sound) waves in gas should be

where, E_T is the isothermal coefficient of volume elasticity and it is equal to the pressure of the gas.

Laplace’s Correction

According to Laplace, the propagation of longitudinal wave is an adiabatic process. Therefore, velocity of longitudinal (sound) wave in gas should be

where, E_S, is the adiabatic coefficient of volume elasticity and it is equal to γ p.

Factors Affecting Velocity of Longitudinal (Sound) Wave

(i) Effect of Pressure The formula for velocity of sound in a gas.

Therefore, (p/ρ) remains constant at constant temperature.

Hence, there is no effect of pressure on velocity of longitudinal wave.

(ii) Effect of Temperature Velocity of longitudinal wave in a gas

Velocity of sound in a gas is directly proportional to the square root of its absolute temperature.

If v₀ and v_t are velocities of sound in air at O°C and t°C, then

(iii) Effect of Density The velocity of sound in gaseous medium

⇒

The velocity of sound in a gas is inversely proportional to the square root of density of the gas.

(iv) Effect of Humidity The velocity of sound increases with increase in humidity in air.

Shock Waves

If speed of a body in air is greater than the speed of sound, then it .s called supersonic speed. Such a body leaves behind it a conical “egion of disturbance which spreads continuously. Such a disturbance is Called a shock wave.

Speed of Transverse Motion

On a stretched string v = √(T/m)

where, T = tension in the string and m = mass per unit length of the string.

Speed of transverse wave in a solid v = √(η/ρ)

where, η is modulus of rigidity and ρ is density of solid.

Plane Progressive Simple Harmonic Wave

Equation of a plane progressive simple harmonic wave

where,

• y = displacement,
• a = amplitude of vibration
• λ = wavelength of wave, of particle,
• T = time period of wave,
• x = distance of particle from the origin and
• u = velocity of wave.

Important Relation Related to Equation of Progressive Wave

Relation between phase difference and path difference and time difference

Echo

The repetition of sound caused by the reflection of sound waves is called an echo.

Sound persists on ear for 0.1 s.

The minimum distance from a sound reflecting surface to hear an echo is 16.5 m.

If first echo be heard after It second, second echo after ~ second, then third echo will be heard after (t₁ + t2)s.

Superposition of Waves

Two or more progressive waves can travel simultaneously in the medium without effecting the motion of one another. Therefore, resultant displacement of each particle of the medium at any instant is equal to vector sum of the displacements produced by two waves separately. This principle is called principle of superposition.

Interference

When two waves of same frequency travel in a medium simultaneously in the same direction, then due to their superposition, the resultant intensity at any point of the medium is different from the sum intensities of the two waves. At some points the intensity of the resultant wave is very large while at some other points it is very small Or zero. This phenomenon is called interference of waves.

Constructive Interference

Phase difference between two waves = 0, 2π, 4π
Maximum amplitude = (a + b)
Intensity ∝ (Amplitude)² ∝ (a + b)²

In general, amplitude = √a² + b² +2abcos φ

Destructive Interference

Phase difference between two waves = π, 3π, 5π

Minimum amplitude = (a ~ b) = Difference of component amplitudes.

Intensity ∝ (Amplitude)²

∝ (a – b)²

A vibrating tuning fork, when rotated near ear, produced loud sound and silence due to constructive and destructive interference.

Beats

When two sound waves of nearly equal frequencies are produced simultaneously, then intensity of the resultant sound produced by their superposition increases and decreases alternately with time. This rise and fall intensity of sound is called beats.

The number of maxima or minima heard in one second is called beats frequency.

[The difference of frequencies should not be more than 10. Sound persists on human ear drums for 0.1 second. Hence, beats will not be heard if the frequency difference exceeds 10]

Number of beats heard per second = n₁ – n₂ = difference of frequencies of two waves.

Maximum amplitude = (a₁ + a₂)

Maximum intensity = (Maximum amplitude)² = (a₁ + a₂)²

For loudness, time intervals are

Stationary or Standing Waves

When two similar waves propagate in a bounded medium in opposite directions, then due to their superposition a new type of wave is obtained, which appears stationary in the medium. This wave is called stationary or standing waves.

Equation of a stationary wave

Nodes and antinodes are obtained alternatively in a stationary waves.

• At nodes, the displacement of the particles remains minimum, strain is maximum,pressure and density variations are maximum.
• At antinodes, the displacement of the particles remains maximum, strain is minimum, pressure and density variations are minimum.
• The distance between two consecutive nodes or two consecutive antinodes = λ/2.
• The distance between a node and adjoining antinode = λ/4.
• All the particles between two nodes vibrate in same phase. particles on two sides of a node vibrate in opposite phase.
• n consecutive nodes are separated n by ((n -l)λ/2) .

Vibrations in a Stretched String

Velocity of a transverse wave in a stretched string.

Where,T is tension in the string and m is mass per unit length of the string.

Fundamental frequency or frequency of first harmonic

Frequency of first overtone or second harmonic

n₂ = 2(v/2l) = 2n₁

Frequency of second overtone or third harmonic

n₃ = 3(v/2l) = 3n₁

n₁ : n₂ : n₃ : … = 1 : 2 : 3 : …

Organ Pipes

Organ pipes are those cylindrical pipes which are used for produe musical (longitudial) sounds. Organ pipes are of two types

1. Open Organ Pipe Cylindrical pipes open at both ends.
2. Closed Organ Pipe Cylindrical pipes open at one end closed at other end.

Fundamental Note

It is the sound of lowest frequency produced in fundamental note., vibration of a system.

Overtones Tones having frequencies greater than the runoamen note are called overtones.

Harmonics When the frequencies of overtone are integral multiples of the fundamental, then they are known as harmonics. Thus note of lowest frequency n is called fundamental note or first harmonics. The note of frequency 2n is called second harmonic or first overtone.

Vibrations in Open Organ Pipe

fundamental frequency or frequency of first harmonic

n₁ = (2v/l)

frequency of first overtone or second harmonic

n₂ = (2v/2l) = 2n₁

Frequency of second overtone or third harmonic

n₃ = (3v/2l) = 3n₁

n₁ : n₂ : n₃ : …. = 1 : 2 : 3 …

Therefore,even and odd harmonics are produced by an open organ pipe.

Vibrations in Closed Organ Pipe

Fundamental frequency or frequency of first harmonic

n₁ = (v/4l)

Frequency of first harmonic or third harmonic

n₃ = 5(v/4l) = 5n₁

n₁ : n₂ : n₃ : … 1 : 3 : 5 : …

Frequency of second harmonic or fifth harmonic

n₃ = (3v/2l) = 3n₁

Therefore only even harmonics are produced by a closed organ pipe.

End Correction

Antinode is not obtained at exact open end but slightly above it. The distance between open and antinode is called end correction.

It is denoted by e.

• Effective length of an open organ pipe = (l + 2e)
• Effective length of a closed organ pipe = (1 + e)
• If r is the radius of organ pipe, then e = 0.6 r

Factors Affecting Frequency of Pipe

1. Length of air column, n ∝ (1/l)
2. Radius of air column, n ∝ (1/r)
3. Temperature of air column, n ∝ √T
4. Pressure of air inside air column, n ∝ √p
5. Density of air, n ∝ (1/√ρ)
6. Velocity of sound in air column, n ∝ v

Resonance Tube

Resonance tube is a closed organ pipe in which length of air coluDf can be changed by changing height of liquid column in it.

Melde’s Experiment 

In longitudinal mode, vibrations of the prongs of tuning fork are the length of the string.

In transverse mode, vibrations of tuning fork are at 90° to the length of string.

In both modes of vibrations, Melde’s law

p²T = constant, is obeyed.

Characteristics of Musical Sound

Musical sound has three characteristics

(i) Intensity or Loudness Intensity of sound is energy transmitted per second per unit area by sound waves. Its SI unit is watt/metre². Intensity is measured in decibel (dB).

(ii) Pitch or Frequency Pitch of sound directly depends on frequency.

A shrill and sharp sound has higher pitch and a grave and dull sound has lower pitch.

(iii) Quality or Timbre Quality is the characteristic of sound that differentiates between two sounds of same intensity and same frequency .

Quality depends on harmonics and their relative order and intensity.

Doppler’s Effect

The phenomena of apparent change in frequency of source due to a relative motion between the source and observer is called Doppler’s effect.

(i) When Source is Moving and Observer is at Rest When source is moving with ‘velocity towards an observer at rest, then apparent frequency

(ii) When Source is at Rest and Observer is Moving When observer is moving with velocity VO’ towards a source at rest, then apparent frequency.

(iii) When Source and Observer Both are Moving

(a) When both are moving in same direction along the direction of propagation of sound, then

Transverse Doppler’s Effect

(i) The Doppler’s effect in sound does not take place in the transverse direction.

(ii) As shown in figure, the position of a source is S and of observer is O. The component of velocity of source towards the observer is V cos θ. For this situation, the approach frequency is

f ‘ which will now be a function of θ so, it will no more constant.

Similarly, if the source is moving away from the observer as shown above, with velocity component V_s cos θ then,

(iii) If θ = 90°, the V_s cos θ = 0 and there is no shift in the frequency.

Thus, at point P, Doppler’s effect does not occur.

Effect of Wind

If wind is also blowing with a velocity w in the direction of sound, then its velocity is added to the velocity of sound. Hence, in this condition the apparent frequency is givenby

Applications of Doppler’s Effect

The measurement of Doppler shift has been used

1. by police to check overspeeding of vehicles.
2. at airports to guide the aircraft.
3. to study heart beats and blood flow in different parts of the body.
4. by astrophysicist to measure the velocities of plants and stars.

A motion which repeats itself identically after a fixed interval of time is called periodic motion. e.g., orbital motion of the earth around the sun, motion of arms of a clock, motion of a simple pendulum etc.

Oscillatory Motion

A periodic motion taking place to and fro or back and forth about a fixed point, is called oscillatory motion, e.g., motion of a simple pendulum, motion of a loaded spring etc.

Note Every oscillatory motion is periodic motion but every periodic motion is not oscillatory motion.

Harmonic Oscillation

The oscillation which can be expressed in terms of single harmonic function, i.e., sine or cosine function, is called harmonic oscillation.

Simple Harmonic Motion

A harmonic oscillation of constant amplitude and of single frequency under a restoring force whose magnitude is proportional to the displacement and always acts towards mean Position is called Simple Harmonic Motion (SHM).

A simple harmonic oscillation can be expressed as

y = a sin ωt

or y = a cos ωt

Where a = amplitude of oscillation.

Non-harmonic Oscillation

A non-harmonic oscillation is a combination of two or more than two harmonic oscillations.

It can be expressed as y = a sin ωt + b sin 2ωt

Some Terms Related to SHM

(i) Time Period Time taken by the body to complete one oscillation is known as time period. It is denoted by T.

(ii) Frequency The number of oscillations completed by the body in one second is called frequency. It is denoted by v.

Its SI unit is ‘hertz’ or ‘second^-1‘.

Frequency = 1 / Time period

(iii) Angular Frequency The product of frequency with factor 2π is called angular frequency. It is denoted by ω.

Angular frequency (ω) = 2πv

Its SI unit is ‘hertz’ or ‘second^-1‘.

(iv) Displacement A physical quantity which changes uniformly with time in a periodic motion. is called displacement. It is denoted by y.

(v) Amplitude The maximum displacement in any direction from mean position is called amplitude. It is denoted by a.

(vi) Phase A physical quantity which express the position and direction of motion of an oscillating particle, is called phase. It is denoted by φ.

Simple harmonic motion is defined as the projection of a uniform circular motion on any diameter of a circle of reference.

Some Important Formulae of SHM

(i) Displacement in SHM at any instant is given by

y = a sin ωt

or y = a cos ωt

where a = amplitude and

ω = angular frequency.

(ii) Velocity of a particle executing SHM at any instant is given by

v = ω √(a² – y²)

At mean position y = 0 and v is maximum

v_max = aω

At extreme position y = a and v is zero.

(iii) Acceleration of a particle executing SHM at any instant is given by

A or α = – ω² y

Negative sign indicates that the direction of acceleration is opposite to the direction in which displacement increases, i.e., towards mean position.

At mean position y = 0 and acceleration is also zero.

At extreme position y = a and acceleration is maximum

A^max = – aω²

(iv) Time period in SHM is given by

T = 2π √Displacement / Acceleration

Graphical Representation

(i) Displacement – Time Graph

(ii) Velocity – Time Graph

(iii) Acceleration – Time Graph

Note The acceleration is maximum at a place where the velocity is minimum and vice – versa.

For a particle executing SlIM. the phase difference between

(i) Instantaneous displacement and instantaneous velocity

= (π / 2) rad

(ii) Instantaneous velocity and instantaneous acceleration

= (π / 2) rad

(iii) Instantaneous acceleration and instantaneous displacement

= π rad

The graph between velocity and displacement for a particle executing SHM is elliptical.

Force in SHM

We know that, the acceleration of body in SlIM is α = -ω² x

Applying the equation of motion F = ma,

We have, F = – mω² x = -kx

Where, ω = √k / m and k = mω² is a constant and sometimes it is called the elastic constant.

In SHM, the force is proportional and opposite to the displacement.

Energy in SHM

The kinetic energy of the particle is K = 1 / 2 mω² (A² – x²)

From this expression we can see that, the kinetic energy is maximum at the centre (x = 0) and zero at the extremes of oscillation (x ± A).

The potential energy of the particle is U = 1 / 2 mω² x²

From this expression we can see that, the potential energy has a minimum value at the centre (x = 0) and increases as the particle approaches either extreme of the oscillation (x ± A).

Total energy can be obtained by adding potential and kinetic energies. Therefore,

E = K + U

= = 1 / 2 mω² (A² – x²) + 1 / 2 mω² x²

= 1 / 2 mω² A²

where A = amplitude

m = mass of particle executing SHM.

ω = angular frequency and

v = frequency

Changes of kinetic and potential energies during oscillations.

The frequency of kinetic energy or potential energy of a particle executing SHM is double than that of the frequency in SHM.

The frequency of total energy of particles executing SHM is zero as total energy in SHM remains constant at all positions.

When a particle of mass m executes SHM with a constant angular frequency (I), then time period of oscillation

T = 2π √Inertia factor / Spring factor

In general, inertia factor = m, (mass of the particle)

Spring factor = k (force constant)

How the different physical quantities (e.g., displacement, velocity, acceleration, kinetic energy etc) vary with time or displacement are listed ahead in tabular form.

Simple Pendulum

A simple pendulum consists of a heavy point mass suspended from a rigid support by means of an elastic inextensible string.

The time period of the simple pendulum is given by :

T = 2π √l / g

where l = effective length of the pendulum and g = acceleration due to gravity.

If the effective length l of simple pendulum is very large and comparable with the radius of earth (R), then its time period is given by

T = 2π √Rl / (l + R)g

For a simple pendulum of infinite length (l >> R)

T = 2π √R / g = 84.6 min

For a simple pendulum of length equal to radius of earth,

T = 2π √R / g = 60 min

If the bob of the simple pendulum is suspended by a metallic wire of length l, having coefficient of linear expansion α, then due to increase in temperature by dθ, then

Effective length l’ = l (1 + α dθ)

Percentage increase in time period

(T’ / T – 1) * 100 = 50 α dθ

When a bob of simple pendulum of density ρ oscillates in a fluid of density ρ_o (ρ_o < p), then time period get increased.

Increased time period T’ = T √ρ / ρ – ρ_o

When simple pendulum is in a horizontally accelerated vehicle, then its time period is given by

T = 2π √1 / √(a² + g²)

where a = horizontal acceleration of the vehicle.

When simple pendulum is in a vehicle sliding down an inclined plane, then its time period is given by

T = 2π √l / g cos θ

Where θ = inclination of plane.

Second’s Pendulum

A simple pendulum having time period of 2 second is called second’s Pendulum.

The effective length of a second’s pendulum is 99.992 em of approximately 1 metre on earth.

Conical Pendulum

If a simple pendulum is fixed at one end and the bob is rotating in a horizontal circle, then it is called a conical pendulum.

In equilibrium T sin θ = mrΩ²

Its time period T = 2π √mr / T sin θ

Compound Pendulum

Any rigid body mounted, so that it is capable of swinging in a vertical plane about some axis passing through it is called a physical or compound pendulum.

Its time period is given by

T = 2π √l / mg l

where, I = moment of inertia of the body about an axis passing through the centre of suspension,

m = mass of the body and

l = distance of centre of gravity from the centre of suspension.

Torsional Pendulum

Time period of torsional pendulum is given by

T = 2π √I / C

where, I = moment of inertia of the body about the axis of rotation and

C = restoring couple per unit twist.

Physical Pendulum

When a rigid body of any shape is capable of oscillating about an axis (mayor may not be passing through it). it constitutes a physical pendulum.

T = 2π √I / mgd

• The simple pendulum whose time period is same as that of a physical pendulum is termed as an equivalent simple pendulum.

T = 2π √I / mgd = 2 π √l / g

• The length of an equivalent simple pendulum is given by l = I / md

Spring Pendulum

A point mass suspended from a massless (or light) spring constitutes a spring pendulum. If the mass is once pulled downwards so as to stretch the spring and then released. the system oscillated up and down about its mean position simple harmonically. Time period and frequency of oscillations are given by

T = 2π √m / k or v = 1 / 2π √k / m

If the spring is not light but has a definite mass m_s, then it can be easily shown that period of oscillation will be

T = 2π √(m + m_s / 3) / k

Oscillations of Liquid in a U – tube

If a liquid is filled up to height h in both limbs of a U-tube and now liquid is depressed upto a small distance y in one limb and then released, then liquid column in U-tube start executing SlIM.

The time period of oscillation is given by

T = 2π √h / g

Oscillations of a floating cylinder in liquid is given by

T = 2π √l / g

where I = length of the cylinder submerged in liquid in equilibrium.

Vibrations of a Loaded Spring

When a spring is compressed or stretched through a small distance y from mean position, a restoring force acts on it.

Restoring force (F) = – ky

where k = force constant of spring.

If a mass m is suspended from a spring then in equilibrium,

mg = kl

This is also called Hooke’s law.

Time period of a loaded spring is given by

T = 2π √m / k

When two springs of force constants k₁ and k₂ are connected in parallel to mass m as shown in figure, then

(i) Effective force constant of the spring combination

k = k₁ + k₂

(ii) Time period T = 2π √m / (k₁ + k₂)

When two springs of force constant k₁ and k₂ are connected in series to mass m as shown in figure, then

(i) Effective force constant of the spring combination

1 / k = 1 / k₁ + 1 / k₂

(ii) Time period T = 2π √m(k₁ + k₂) / k₁k₂

Free Oscillations

When a body which can oscillate about its mean position is displaced from mean position and then released, it oscillates about its mean position. These oscillations are called free oscillations and the frequency of oscillations is called natural frequency.

Damped Oscillations

Oscillations with a decreasing amplitude with time are called damped oscillations.

The displacement of the damped oscillator at an instant t is given by

x = x_oe^{– bt / 2m} cos (ω’ t + φ)

where x_oe^{– bt / 2m} is the amplitude of oscillator which decreases continuously with time t and ω’.

The mechanical energy E of the damped oscillator at an instant t is given by

E = 1 / 2 kx²_oe^{– bt / 2m}

Un-damped Oscillations

Oscillations with a constant amplitude with time are called un-damped oscillations.

Forced Oscillations

Oscillations of any object with a frequency different from its natural frequency under a periodic external force are called forced oscillations.

Resonant Oscillations

When an external force is applied on a body whose frequency is an integer multiple of the natural frequency of the body, then its amplitude of oscillation increases and these oscillations are called resonant oscillations.

Lissajous’ Figures

If two SHMs are acting in mutually perpendicular directions, then due to then: superpositions the resultant motion, in general, is a curvelloop. The shape of the curve depends on the frequency ratio of two SHMs and initial phase difference between them. Such figures are called Lissajous’ figures.

1. Let two SHMs be of same frequency (e.g., x = a₁ sinωt and y = a₂ sin (omega;t + φ), then the general equation of resultant motion is found to be

x² / a²₁ + y² / b²₂ – 2xy / a₁a_{2/sub> cos φ = sin² φ}

The equation represents an ellipse. However, if φ = O° or π or nπ, then the resultant curve is a straight inclined line.

2. Let two SHMs be having frequencies in the ratio 1 : 2, then, in general, the Lissajous figure is a figure of eight (8).

There are three methods of heat transmission

(i) Conduction In solids, heat is transmitted from temperature to lower temperature without actual movements the particles. This mode of transmission of heat. is conduction.

(ii) Convection The process of heat-transmission in which particles of the fluid (liquid or gas) move. is called convection. Land breeze and see-breeze are formed due to convection.

(iii) Radiation The process of heat transmission in the form electromagnetic waves, is called radiation.

Radiation do not require any medium for propagation.

It propagates without heating the intervening medium.

The heat energy transferred by radiation, is called energy.

Heat from the sun reaches the earth by radiation.

Thermal Conductivity

In solids, heat is transferred through conduction. We will conduction of heat through a solid bar.

Conduction of Heat in a Conducting Rod

Steady State

The state of a conducting rod in which no part of the rod absorbs nea is caned the steady state.

Isothermal Surface

A surface of a material whose all points are at the same temperature, is called an isothermal surface.

Temperature Gradient

The rate of change of temperature with distance between two isothermal surfaces is called temperature gradient.

Temperature gradient = Change in temperature / Perpendicular distance = – Δθ / Δx

Its SI unit is ‘^oC per meter’ and dimension is [L^-1θ].

The amount of heat flow in a conducting rod

Q = KA Δθt / l

where K = coefficient of thermal conductivity.
A = area of cross-section,
l = length of rod,
Δθ = temperature difference between the ends of the rod and
t = time.

The SI unit of K is ‘Wm^-1 K^-1‘ and its dimension is [MLT^-3θ^-1].

The value of K is large for good conductors and very small for insulators.

Thermal resistance is given by

R = Δθ / H = l / KA

where, Δθ is temperature difference at the ends of the rod and H rate of flow of heat.

Its SI unit is K/W and its dimensional formula is [ML² T^-3θ^-1].

When Conducting Rods are Connected in Series

The amount of heat flow

Equivalent thermal conductivity

When Two Conducting Rods are Connected in Series

Rate of heat flow

Temperature of contact surface

Equivalent thermal conductivity H = l₁ + l₂ / l₁ / K₁ + l₂ / k₂

Equivalent thermal resistance R = R₁ + R₂

When Conducting Rods are Connected in Parallel

Rate of heat flow

Equivalent thermal conductivity

K = K₁ A₁ + K₂ A₂ / A₁ + A₂

Equivalent thermal resistance

1 / R = 1 / R₁ + 1 / R₂

Ingen-Hausz Experiment

The thermal conductivities of different materials are proportional to the square of the lengths of the melted wax on the rods of these materials in the steady state.

If l₁, l₂, l₃,…. are the lengths of the melted wax on the rods of different materials having coefficient of thermal conductivities K₁, K₂, K₃,…, then

K₁ : K₂ : K₃ :…. = l₁² : l₂² : l₃² :….

or K₁ / K₂ = l₁² / l₂²

Formation of Ice on Lakes

Time taken to form x thickness of ice on lake

t = (ρL / 2Kθ) x²

where ρ density of ice, L = latent heat of freezing of ice.

K = coefficient of thermal conductivity of ice and

θ = temperature above lake.

Time taken to increase the thickness of ice from x₁ to x₂

t = (ρ L / 2Kθ) (x²₂ – x²₁)

Reflectance or Reflecting Power

The ratio of the amount of thermal radiations reflected by a body in a given time to the total amount of thermal radiations incident on the body in that time is called reflectance or reflecting power of the body.

It is denoted by r.

Absorptance or Absorbing Power

The ratio of the amount of thermal radiations absorbed by a body in a given time to the total amount of thermal radiations incident on the body in that time, is called absorptance or absorbing power of the body.

It is denoted by a.

Transmittance or Transmitting Power

The ratio of the amount of thermal radiations transmitted by the body in a given time to the total amount of thermal radiations incident on the body in that time, is called transmittance or transmitting power of the body.

It is denoted by t.

Relation among reflecting power, absorbing power and transmitting power

r + a + t = l

If body does not transmit any heat radiations, then t = 0

∴ r + a = 1

(i) r, a and t all are the pure ratio, so they have no unit dimension.

(ii) For perfect reflector, r = 1, a = 0 and t = O.

(iii) For perfect absorber, a = 1, r = 0 and t = 0 (perfect black body).

(iv) For perfect transmitter, t = 1, a = 0 and r = O.

Emissive Power

Emissive power of a body at a particular temperature is the total amount of thermal energy emitted per unit time per unit area of thel body for all possible wavelengths.

It is denoted by e_λ

Its SI unit is ‘joule sec^-l metre^-2 or ‘watt-metre^-2‘.

Its dimensional formula is [MT^-3].

Emissivity

Emissivity of a body at a given temperature is equal to the ratio of the total emissive power of the body (e_λ) to the total emissive power of a perfectly black body (E_λ) at that temperature.

Emissivity ε = e_λ / E_λ

Perfectly Black Body

A body which absorbs completely the radiations of all wavelengths incident on it, is called a perfectly black body.

For a perfectly black body, emissive power (E_λ) = 1

An ideal black body need not be black in colour.

The radiation from a black body depend upon its temperature only.

These heat radiations do not depend on density mass, size or the nature of the body.

Lamp black is 96% black and platinum black is about 98% black.

A perfectly black body cannot be realised in practice. The nearest example of an ideal black body is the Ferry’s black body.

Kirchhoff’s Law

The ratio of emissive power (e_λ) to the absorptive power (a_λ) cOrresponding to a particular wavelength and at any given temperature is always a constant for all bodies and it is equal to the emissive power (E_λ) of a perfectly black body at the same temperature and corresponding to the same wavelength.

Mathematically e_λ / a_λ = constant (E_λ)

Fraunhofer Lines

These are typical spectral absorption lines. These are dark but not perfectly black due to their lower intensity as compared to the remaining part of spectrum.

These lines are produced when a cold gas is in between a broad spectrum photon source and detector.

A set of several hundred dark lines appearing against the bright back ground of the continuous solar spectrum and produced by absorption of light by the cooler gases in the sun’s outer atmosphere at frequencies corresponding to the atomic transition frequencies of these gases.

About 20000 such dark lines have been detected so far. These dark lines belong to hydrogen, helium, sodium, iron, calcium etc.

Kirchhoff’s law explains this phenomenon.

Moon covers the photosphere the central part of sun at the time of total solar eclipse. Thus, the elements present in the chromosphere emit certain wavelength which they had absorbed. Because of this reason the Fraunhofer lines appears as bright lines at the time of total solar eclipse.

Variation of Colour with Temperature

(i) At temperature 525°e, the colour of object is red.

(ii) At temperature 900o e, the colour of object is cherry red.

(iii) At temperature 1100o e, the colour of object is orange red.

(iv) At temperature l200o e, the colour of object is yellow.

(v) At temperature l600o e, the colour of object is white.

Stefan’s Law

Heat energy emitted per second per unit area of a perfectly black body

E ∝ ⇒T⁴ ⇒ E = σ T⁴

where σ is Stefan’s constant and its value is 5.67 x 10^-8 Wm^-2 K^-4.

If T_o is the temperature of the surroundings, then

E = σ (T⁴ – T⁴_o)

If e is the emissivity of the body, then

E = ε σ T⁴

Energy radiated by whole body in t time,

E = σAt T⁴

Newton’s Law of Cooling

The rate of loss of heat of a liquid is directly proportional to difference in temperature of the liquid and its surroundings, i.e.

dT / dt = E ∝ (T – T_o)

where T and T_o are the temperatures of the liquid and surroundings.

Wien’s Displacement Law

For black body radiation the rate of energy radiation per unit area per unit wavelength range at constant kelvin temperature can be against wavelength.

λ_m T = constant (b)

where λ_m = wavelength corresponding to which maximum energy is radiated,

T = absolute temperature and

b = Wien’s constant = 2.898 x 10^-3 = 3 X 10^-3 mK

When temperature of a black body increases, colour changes towards higher frequency, i.e., from red → orange → yellow → green → blue → violet.

Solar Constant

The amount of heat received from the sun by one square centimetre area of a surface placed normally to the sun rays at mean distance the earth from the sun is known as solar constant. It is denoted by S.

S = (r / R)² σ T⁴

Here, r is the radius of sun and R is the mean distance of earth from the centre of sun. Value of solar constant is 1.937 cal cm^-2 min^-1.

Temperature of violet star is maximum while temperature of red is minimum.

A thermodynamical system is said to be in thermal equilibrium when macroscopic variables (like pressure, volume, temperature, mass, composition etc) that characterise the system do not change with time.

Thermodynamical System

An assembly of an extremely large number of particles whose state can be expressed in terms of pressure, volume and temperature, is called thermodynamic system.

Thermodynamic system is classified into the following three systems

(i) Open System It exchange both energy and matter with surrounding.

(ii) Closed System It exchanges only energy (not matter) with surroundings.

(iii) Isolated System It exchanges neither energy nor matter with the surrounding.

A thermodynamic system is not always in equilibrium. For example, a gas allowed to expand freely against vacuum. Similary, a mixture of petrol vapour and air, when ignited by a spark is not an equilibrium state. Equilibrium is acquired eventually with time.

Thermodynamic Parameters or Coordinates or Variables

The state of thermodynamic system can be described by specifying pressure, volume, temperature, internal energy and number of moles, etc. These are called thermodynamic parameters or coordinates or variables.

Work done by a thermodynamic system is given by

W = p * ΔV

where p = pressure and ΔV = change in volume.

Work done by a thermodynamic system is equal to the area enclosed between the p-V curve and the volume axis

Work done in process A-B = area ABCDA

Work done by a thermodynamic system depends not only upon the initial and final states of the system but also depend upon the path followed in the process.

Work done by the Thermodynamic System is taken as

Positive → 4 as volume increases.

Negative → 4 as volume decreases.

Internal Energy (U)

The total energy possessed by any system due to molecular motion and molecular configuration, is called its internal energy.

Internal energy of a thermodynamic system depends on temperature. It is the characteristic property of the state of the system.

Zeroth Law of Thermodynamics

According to this law, two systems in thermal equilibrium with a third system separately are in thermal equilibrium with each other. Thus, if A and B are separately in equilibrium with C, that is if T_A = T_C and T_B = T_C, then this implies that T_A = T_B i.e., the systems A and B are also in thermal equilibrium.

First Law of Thermodynamics

Heat given to a thermodynamic system (ΔQ) is partially utilized in doing work (ΔW) against the surrounding and the remaining part increases the internal energy (ΔU) of the system.

Therefore, ΔQ = ΔU + ΔW

First law of thermodynamics is a restatement of the principle conservation of energy.

In isothermal process, change in internal energy is zero (ΔU = 0).

Therefore, ΔQ = ΔW

In adiabatic process, no exchange of heat takes place, i.e., Δθ = O.

Therefore, ΔU = – ΔW

In adiabatic process, if gas expands, its internal energy and hence, temperature decreases and vice-versa.

In isochoric process, work done is zero, i.e., ΔW = 0, therefore

ΔQ = ΔU

Thermodynamic Processes

A thermodynamical process is said to take place when some changes’ occur in the state of a thermodynamic system i.e., the therrnodynamie parameters of the system change with time.

(i) Isothermal Process A process taking place in a thermodynamic system at constant temperature is called an isothermal process.

Isothermal processes are very slow processes.

These process follows Boyle’s law, according to which

pV = constant

From dU = nC_vdT as dT = 0 so dU = 0, i.e., internal energy is constant.

From first law of thermodynamic dQ = dW, i.e., heat given to the system is equal to the work done by system surroundings.

Work done W = 2.3026μRT l0g₁₀(V_f / V_i) = 2.3026μRT l0g₁₀(p_i / p_f)

where, μ = number of moles, R = ideal gas constant, T = absolute temperature and V_i V_f and P_i, P_f are initial volumes and pressures.

After differentiating P V = constant, we have

i.e., bulk modulus of gas in isothermal process, β = p.

P – V curve for this persons is a rectangular hyperbola

Examples

(a) Melting process is an isothermal change, because temperature of a substance remains constant during melting.

(b) Boiling process is also an isothermal operation.

(ii) Adiabatic Process A process taking place in a thermodynamic system for which there is no exchange of heat between the system and its surroundings.

Adiabatic processes are very fast processes.

These process follows Poisson’s law, according to which

From dQ = nCdT, C_adi = 0 as dQ = 0, i.e., molar heat capacity for adiabatic process is zero.

From first law, dU = – dW, i.e., work done by the system is equal to decrease in internal energy. When a system expands adiabatically, work done is positive and hence internal energy decrease, i.e., the system cools down and vice-versa.

Work done in an adiabatic process is

where T_i and T_f are initial and final temperatures. Examples

(a) Sudden compression or expansion of a gas in a container with perfectly non-conducting wall.

(b) Sudden bursting of the tube of a bicycle tyre.

(c) Propagation of sound waves in air and other gases.

(iii) Isobaric Process A process taking place in a thermodynamic system at constant pressure is called an isobaric process.

Molar heat capacity of the process is C_p and dQ = nC_pdT.

Internal energy dU = nC_v dT

From the first law of thermodynamics
dQ = dU + dW
dW = pdV = nRdT

Process equation is V / T = constant.

p- V curve is a straight line parallel to volume axis.

(iv) Isochoric Process A process taking place in a tlaermodynars system at constant volume is called an isochoric process.

dQ = nC_vdT, molar heat capacity for isochoric process is C_v.

Volume is constant, so dW = 0,

Process equation is p / T = constant

p- V curve is a straight line parallel to pressure axis.

(v) Cyclic Process When a thermodynamic system returns to . initial state after passing through several states, then it is called cyclic process.
Efficiency of the cycle is given by

Work done by the cycle can be computed from area enclosed cycle on p- V curve.

Isothermal and Adiabatic Curves

The graph drawn between the pressure p and the volume V of a given mass of a gas for an isothermal process is called isothermal curve and for an adiabatic process it is called adiabatic curve .

The slope of the adiabatic curve

= γ x the slope of the isothermal curve

Volume Elasticities of Gases

There are two types of volume elasticities of gases

(i) Isothermal modulus of elasticity E_S = p

(ii) Adiabatic modulus of elasticity E_T = γ p

Ratio between isothermal and adiabatic modulus

E_S / E_T = γ = C_p / C_V

where C_p and C_v are specific heats of gas at constant pressure and at constant volume.

For an isothermal process Δt = 0, therefore specific heat,

c = Δ θ / m Δt = &infi;

For an adiabatic process 119= 0, therefore specific heat,

c = 0 / m Δt = 0

Second Law of Thermodynamics

The second law of thermodynamics gives a fundamental limitation to the efficiency of a heat engine and the coefficient of performance of a refrigerator. It says that efficiency of a heat engine can never be unity (or 100%). This implies that heat released to the cold reservoir can never be made zero.

Kelvin’s Statement

It is impossible to obtain a continuous supply of work from a body by cooling it to a temperature below the coldest of its surroundings.

Clausius’ Statement

It is impossible to transfer heat from a lower temperature body to a higher temperature body without use of an extemal agency.

Planck’s Statement

It is impossible to construct a heat engine that will convert heat completely into work.

All these statements are equivalent as one can be obtained from the other.

Entropy

Entropy is a physical quantity that remains constant during a reversible adiabatic change.

Change in entropy is given by dS = δQ / T

Where, δQ = heat supplied to the system

and T = absolute temperature.

Entropy of a system never decreases, i.e., dS ≥ o.

Entropy of a system increases in an irreversible process

Heat Engine

A heat energy engine is a device which converts heat energy into mechanical energy.

A heat engine consists of three parts

(i) Source of heat at higher temperature

(ii) Working substance

(iii) Sink of heat at lower temperature

Thermal efficiency of a heat engine is given by

where Q₁ is heat absorbed from the source,

Q₂ is heat rejected to the sink and T₁ and T₂ are temperatures of source and sink.

Heat engine are of two types

(i) External Combustion Engine In this engine fuel is burnt a chamber outside the main body of the engine. e.g., steam engine. In practical life thermal efficiency of a steam engine varies from 12% to 16%.

(ii) Internal Combustion Engine In this engine. fuel is burnt inside the main body of the engine. e.g., petrol and diesel engine. In practical life thermal efficiency of a petrol engine is 26% and a diesel engine is 40%.

Carnot’s Cycle

Carnot devised an ideal cycle of operation for a heat engine, called Carnot’s cycle.

A Carnot’s cycle contains the following four processes

(i) Isothermal expansion (AB)

(ii) Adiabatic expansion (BO)

(iii) Isothermal compression (CD)

(iv) Adiabatic compression (DA)

The net work done per cycle by the engine is numerically equal to the area of the loop representing the Carnot’s cycle .

After doing the calculations for different processes we can show that

[Efficiency of Carnot engine is maximum (not 1000/0) for given temperatures T₁ and T₂. But still Carnot engine is not a practical
engine because many ideal situations have been assumed while designing this engine which can practically not be obtained.]

Refrigerator or Heat Pump

A refrigerator or heat pump is a device used for cooling things. It absorb heat from sink at lower temperature and reject a larger amount of heat to source at higher temperature.

Coefficient of performance of refrigerator is given by

where Q₂ is heat absorbed from the sink, Q₁ is heat rejected to source and T₁ and T₂ are temperatures of source and sink.

Relation between efficiency (η) and coefficient of performance (β)