CBSE Class 12 Physics Electromagnetic Induction Exam Notes

CBSE Class 12 Physics Electromagnetic Induction Exam Notes. Please refer to the examination notes which you can use for preparing and revising for exams. These notes will help you to revise the concepts quickly and get good marks.

These Class 12 Physics exam notes on Electromagnetic Induction cover magnetic flux, Faraday's and Lenz's laws, induced charge, motional e.m.f. in a straight rod, e.m.f. due to rotation of a rod and a coil, self inductance, energy stored in an inductor, mutual inductance, inductors in series and parallel, L-C oscillations, and the growth and decay of current in an L-R circuit, with the main formulae and derivations.

MAGNETIC FLUX

Magnetic flux (φ) through an area ds in magnetic field B is defined as
φ = B · ds    (i)

Physically it represents the total lines of induction passing through a given area.

[Figure: A plane area with area vector ds and a magnetic field B drawn at an angle θ to the area vector]

Equation (i) can be written as
φ = B ds cos θ    (ii)

where θ is the angle between B and the area vector. According to equation (ii), flux can change not only due to magnetic field and area but also due to the orientation of the area with respect to B.

Dimensional formula of flux is [ML²T⁻²A⁻¹].

SI unit of magnetic flux is volt-sec, which is also known as weber.

As 1 volt = 10⁸ emu of potential, the C.G.S. unit of flux, maxwell (Mx), is related to weber through the relation
1 wb = 1 V × s = 10⁸ emu of potential × s = 10⁸ Mx.

ELECTROMAGNETIC INDUCTION

If a magnet passes through a coil in a closed circuit containing a galvanometer, an electric current is produced in the circuit. The direction of the current so induced in the circuit is reversed when the magnet recedes away from the coil. The current so produced lasts so long as there is relative motion between the magnet and the coil. It is shown that whenever the magnetic flux linked with a closed circuit changes, an induced e.m.f. is produced in the circuit and lasts as long as the flux changes. Such currents are produced due to induced electromotive force and the phenomenon is called electromagnetic induction. The magnitude and direction of induced electromagnetic force is given by the following Faraday's and Lenz's laws respectively.

[Figure: A coil connected in a closed circuit with a galvanometer G, and a bar magnet with poles marked S and N placed near the coil]

Faraday's First Law: Whenever the magnetic flux linked with a closed circuit changes, an induced electromotive force is produced which produces an induced current in the circuit which lasts as long as the change lasts.

Faraday's Second Law: The induced e.m.f. is equal to the negative of the rate of change of flux through the circuit.
e = -\( \frac{dφ}{dt} \)

The negative sign shows that the induced e.m.f. opposes the changes in the magnetic flux.

If the coil has N number of turns, then e = -\( \frac{N dφ}{dt} \).

Lenz's Law: The direction of induced electromotive force is such that it opposes the cause that produces the electromagnetic induction.

If the magnetic flux changes from φ₁ to φ₂ in time t, the average induced e.m.f. is given by
e (avg) = -\( \frac{N(φ_2 - φ_1)}{t} \).

When the magnetic flux φ through a closed circuit of known resistance R changes, the quantity of induced charge q can be found as below:
As e = -N\( \frac{Δφ}{Δt} \), i = \( \frac{e}{R} \) = \( \frac{N}{R} \)\( \frac{Δφ}{Δt} \)
q = i Δt = \( \frac{N}{R} \)\( \frac{Δφ}{Δt} \) Δt = \( \frac{N Δφ}{R} \) = \( \frac{\text{Total change of flux}}{\text{Resistance}} \)

INDUCTION DUE TO MOTION OF A STRAIGHT ROD IN THE MAGNETIC FIELD

Consider a straight conducting rod CD moving with velocity v towards right along a U shaped conductor in a uniform magnetic field B directed into the page. The motion of the conductor CD results in changing the area from CDEF to C′D′EF. It results in a change of area CDD′C′ in the magnetic flux producing an increase in the magnetic flux dφ as
dφ = B · A

[Figure: A U-shaped conductor CFED in a magnetic field directed into the page (crosses), with the rod CD of length l moving to the right to the position C′D′]

If l is the length of rod CD, which moves with velocity v in time dt, the change in area perpendicular to the field = CDD′C′ = l v dt
∴ dφ = B l v dt

The magnitude of induced e.m.f., e = \( \frac{dφ}{dt} \) = B l v

If R is the resistance of the loop, the induced current is
I = \( \frac{B l v}{R} \)

The direction of the induced current is given by Fleming's Right hand rule.

[Fleming's Right Hand Rule: It states that if the thumb and the first two fingers of the right hand are stretched mutually perpendicular to each other and if the forefinger gives the direction of the magnetic field and the thumb gives the direction of motion of the conductor, then the central finger gives the direction of the induced current.]

The current in the above mentioned loop is in the anticlockwise direction. If the loop CDEF is moved towards right with velocity v, the induced current I will be flowing in the clockwise direction and this current will produce forces F₁ and F₂ on arms CF and DE respectively which being equal and opposite will cancel.

[Figure: A rectangular loop CDEF moving to the right with velocity v in a field directed into the page (crosses); forces F₁ on arm CF (upward), F₂ on arm DE (downward) and F₃ on arm CD of length l (to the left)]

Force F₃ on arm CD = B I l where CD = l
∴ F₃ = B l \( \left(\frac{B l v}{R}\right) \) = \( \frac{B^{2} l^{2} v}{R} \), where R is the resistance of the closed loop.

Power to pull the loop = F₃ v = \( \frac{B^{2} l^{2} v^{2}}{R} \).

This work is completely converted to heat due to the current flowing in the loop; the heat produced in the loop = I²R = \( \frac{B^{2} l^{2} v^{2}}{R} \).

E.M.F. INDUCED DUE TO ROTATION IN MAGNETIC FIELD

(i) Rod rotating in a magnetic field: If a linearly conducting rod of length l moves with a velocity v perpendicular to a magnetic field B, the induced e.m.f. = E₀ = B l v.

If the rod of length l is rotating in a magnetic field with angular velocity ω,
Velocity of different parts is different and increases moving from O → A.
Velocity of element at a distance x from O is ωx.
Induced emf across element of length dx.
| de | = (dx) · (ωx) · B
= ωB · (x dx)

[Figure: A rod OA of length L rotating about O in a uniform magnetic field directed into the page (crosses), with a small element dx marked at distance x from O and the angle θ shown]

⇒ ∫ de = ωB ∫ x dx, with x from 0 to L
= ωB \( \left(\frac{L^{2}}{2}\right) \)
⇒ | e | = \( \frac{1}{2} \) B ω L²

Direction of emf is given by right hand thumb rule.

(ii) Coil rotating in a magnetic field: If a rectangular conducting coil of area A and N turns is rotated in a uniform magnetic field B with angular velocity ω, as shown in the figure.

[Figure: A rectangular coil rotating with angular velocity ω about a vertical axis in a uniform horizontal magnetic field B]

As the coil rotates, an induced e.m.f., E, is produced due to change of flux. At any instant the area vector of the coil makes an angle θ with the magnetic field, and the flux linked with the coil is
φ = BA cos θ where θ = ωt
⇒ φ = BA cos ωt
\( \frac{dφ}{dt} \) = -BAω sin ωt
Using Faraday's law
e = BAω sin ωt
or e = e₀ sin ωt

The induced e.m.f. has a sinusoidal variation with time and has a maximum value of e₀ = NBAω.

Such a coil converts mechanical energy into electrical energy. It provides the basic principle on which an alternating current (A.C.) generator is based.

(iii) Change of area inside magnetic field changes: Let a rectangular coil of width L and length x inside the magnetic field, flux linked with coil,
φ = BA
= (BL · x)
\( \frac{dφ}{dt} \) = BL \( \frac{dx}{dt} \) = (BLv)

[Figure: A rectangular coil ABCD of width L and length x in a field directed into the page (crosses), moving to the right with velocity v]

According to Faraday's law
e = -LBv
or | e | = LvB
The direction of induced emf is given by Lenz's law.

(iv) Flux linked with a coil also changes when the magnetic field over the coil changes with time.

INDUCTANCE

Self Inductance: A magnetic flux is linked with a coil when a constant current is passed through it. However, when the current is increased or reduced in the coil, it results in a change of magnetic flux due to which an e.m.f. is induced in the coil and is called self induced e.m.f. due to the phenomenon of self induction.

If a current I is flowing in a coil, a magnetic flux φ is linked to the coil which is directly proportional to the current
∴ φ ∝ I or φ = LI

Where L is a constant of proportionality and is called the self inductance of the coil or simply the inductance of the coil.
∴ E.M.F. induced in the coil, E = -\( \frac{dφ}{dt} \) = -L \( \frac{dI}{dt} \).

The self inductance of a coil is the e.m.f. induced in it when the rate of change of current is unity. The unit of inductance is Henry. One Henry is defined as the inductance of a coil in which an e.m.f. of 1 volt is produced when the current in the coil is changing at the rate of one ampere per second. If a solenoid has n number of turns per meter and l is its length with total number of turns N = nl and area of cross section A, its inductance L is
L = μ₀ n² A l = \( \frac{μ_0 N^{2} A}{l} \).

ENERGY STORED IN AN INDUCTOR

The energy of a capacitor is stored in the electric field between its plates. Similarly an inductor has the capability of storing energy in its magnetic field.

A changing current in an inductor causes an emf between its terminals. The work done per unit time is power.
P = \( \frac{dW}{dt} \) = -ei = -Li \( \frac{di}{dt} \)

From dW = -dU or \( \frac{dW}{dt} \) = -\( \frac{dU}{dt} \)
we have, \( \frac{dU}{dt} \) = Li \( \frac{di}{dt} \)
or dU = Li di

The total energy U supplied while the current increases from zero to a final value i is
U = L ∫ i di (from 0 to i) = \( \frac{1}{2} \) L i²
U = \( \frac{1}{2} \) L i²

This is the expression for the energy stored in the magnetic field of an inductor when a current i flows through it. The source of this energy is the external source of emf that supplies the current.

MUTUAL INDUCTANCE

If an inductor is placed close to another inductor, an increase or decrease of the current in the first inductor produces a magnetic flux linked with this coil which produces an induced e.m.f. in the second coil and is called mutual induction. If a current I_P is flowing at any time in a primary coil P, then the magnetic flux linked with the secondary coil S is φ_s which is directly proportional to I_P.
φ_s ∝ I_P, φ_s = M I_P

Where M is a constant called the mutual inductance of the two coils.

[Figure: A primary coil P connected to a battery and a resistor carrying current I_P, placed close to a secondary coil S which is connected to a voltmeter v]

Induced e.m.f. in the secondary coil = e = -\( \frac{dφ_s}{dt} \)
∴ e = -M \( \frac{dI_P}{dt} \).

The mutual inductance M of two coils or circuits having self inductance L₁ and L₂ is given by
M = K √(L₁L₂)    (1)

where K is a constant called the "co-efficient of coupling". If coils are wound over each other, the coupling is said to be tight, otherwise loose. For tight coupling K = 1 so M = √(L₁L₂) while for other coupling 0 < K < 1, so M < √(L₁L₂).

Furthermore, from expression (1) it is clear that if L = 0, M will be zero. This means a system can't have mutual inductance without having self inductances; however the converse may not be true, i.e., mutual inductance of coils is zero when there is no coupling.

COILS IN SERIES AND PARALLEL

(i) Coils in series: Using Kirchhoff's second law in the loop
E - L₁ \( \frac{dI}{dt} \) - L₂ \( \frac{dI}{dt} \) = 0
⇒ E - (L₁ + L₂) \( \frac{dI}{dt} \) = 0
or E - L \( \frac{dI}{dt} \) = 0
where L is the equivalent self inductance of the circuit, and
L = L₁ + L₂

[Figure: Two inductors L₁ and L₂ in series with a battery E, the induced e.m.f.s e₁ and e₂ marked across them and the current i shown]

(ii) Coils in parallel: Using Kirchhoff's second law in the loop I
E - L₁ \( \frac{dI_1}{dt} \) = 0
or E = L₁ \( \frac{dI_1}{dt} \)
also E = L₂ \( \frac{dI_2}{dt} \) (using Kirchhoff's second law in loop II)
⇒ E \( \left(\frac{1}{L_1} + \frac{1}{L_2}\right) \) = \( \frac{d}{dt} \)(I₁ + I₂)
or \( \frac{E}{L} \) = \( \frac{dI}{dt} \)
where L is the equivalent inductance and \( \frac{1}{L} \) = \( \frac{1}{L_1} \) + \( \frac{1}{L_2} \).

[Figure: Inductors L₁ and L₂ connected in parallel across a battery E, with the currents I₁, I₂ and I marked and the two loops labelled I and II]

In case of series grouping of two inductors, if mutual inductance is also taken into account with K = 1 then,
L = L₁ + L₂ ± 2M

[Figure: Three series arrangements of inductors L₁ and L₂ labelled L = L₁ + L₂ + 2M, L = L₁ + L₂ - 2M and L = L₁ + L₂ - 2M]

L - C OSCILLATIONS

If an inductor of inductance L is connected to a capacitor of capacitance C as shown in the circuit, there is a current I flowing through the circuit at any time during the discharge of the capacitor.

[Figure: A charged capacitor C connected in a loop with an inductor L, with the current I marked in the loop]

The potential difference across the capacitor at that time = \( \frac{Q}{C} \), where Q is the charge on the capacitor at any time, and at t = 0 the charge is Q₀.

Using Kirchhoff's law in the loop
-\( \frac{Q}{C} \) - L \( \frac{dI}{dt} \) = 0 ⇒ \( \frac{Q}{C} \) = -L \( \frac{dI}{dt} \)
As I = -\( \frac{dQ}{dt} \), \( \frac{Q}{C} \) = -L \( \frac{d^{2}Q}{dt^{2}} \)
\( \frac{d^{2}Q}{dt^{2}} \) + \( \frac{Q}{LC} \) = 0
\( \frac{d^{2}Q}{dt^{2}} \) + ω² Q = 0,

The solution of this differential equation gives variation of Q sinusoidally as Q = Q₀ cos ωt (assuming that the capacitor has charge Q₀ at t = 0).

Where ω = \( \frac{1}{\sqrt{LC}} \).
I = -\( \frac{dQ}{dt} \) = Q₀ ω sin ωt
or I = Q₀ ω sin(ωt) = I₀ sin(ωt)

Energy stored in the capacitor at any time t is
U₁ = \( \frac{1}{2} \) · \( \frac{q^{2}}{C} \) = \( \frac{Q_0^{2}}{2C} \) cos² ωt
or U₁ = \( \frac{Q_0^{2}}{4C} \) (1 + cos 2ωt)

This shows that the energy in the capacitor also changes sinusoidally with angular frequency 2ω.

Energy stored in the inductor at any time t:
U₂ = \( \frac{1}{2} \) L I²
= \( \frac{1}{2} \) L I₀² sin² ωt
= \( \frac{L I_0^{2}}{4} \) (1 - cos 2ωt)
= \( \frac{L I_0^{2}}{4} \) [1 + cos(2ωt - π/2)]

U₂ also changes sinusoidally with angular frequency 2ω but lags by π/2 behind U₁.

GROWTH AND DECAY OF CURRENT IN L - R CIRCUIT

Consider an inductor of inductance L connected in series with a resistor R which is connected in series to a battery of e.m.f. E through a two way key A, B, S as shown in the circuit diagram. When the switch S is connected to A, the current in the circuit grows from zero value. When the current starts growing through the inductance, a back e.m.f. is induced in the coil due to self induction which opposes the rate of growth of current in the circuit. Similarly, when the switch S is connected to B by disconnecting the battery, the current begins to fall. The current, however, does not fall to zero instantaneously due to the e.m.f. induced in the coil due to self induction which opposes and reduces the rate of decay of current in the circuit.

[Figure: Circuit with an inductor L and a resistor R in series with a battery E and a two way key S that can be connected to point A (battery in circuit) or point B (battery removed)]

Growth of Current: If S is connected to A during the growth of current, let I be the instantaneous current at any time in the circuit. A back e.m.f. equal to L \( \frac{dI}{dt} \) will develop in the circuit so that the effective e.m.f. in the circuit is E - L \( \frac{dI}{dt} \), which is equal to the potential drop of IR across the resistor.
∴ E - L \( \frac{dI}{dt} \) = IR
or E - RI = L \( \frac{dI}{dt} \)
or \( \frac{dI}{E - RI} \) = \( \frac{dt}{L} \)

Integrating this equation between the limits when the current is zero at time t = 0 to the instantaneous current I at time t,
∫ \( \frac{dI}{E - RI} \) (from 0 to I) = ∫ \( \frac{dt}{L} \) (from 0 to t),    I = \( \frac{E}{R} \) [1 - e^-Rt/L]

If I₀ is the maximum current, so that I₀ = \( \frac{E}{R} \)
I = I₀ when exp(-Rt/L) = 0 or t = ∞

Thus the current I approaches a value I₀ asymptotically and grows exponentially to a value equal to E/R. The curve for growth of current in L-R circuit is shown in the figure.

[Figure: Graph of current I against time t for growth of current. The curve rises from zero and approaches a dotted horizontal line at E/R = I₀. The value 0.63 I₀ is marked on the I axis, with a dotted line to the curve at time τ on the t axis]

When t = \( \frac{L}{R} \),
I = I₀ [1 - e^{-(R/L) × (L/R)}]
= I₀ [1 - \( \frac{1}{e} \)] = I₀ [\( \frac{e - 1}{e} \)]
= I₀ [\( \frac{2.718 - 1}{2.718} \)] = 0.63 I₀

The current reaches a value which is equal to 63% of the maximum value I₀ after a time of τ = \( \frac{L}{R} \) second from the beginning.
∴ Time constant of the circuit = τ = \( \frac{L}{R} \).

The time constant τ of a circuit is the time during which the current rises from zero to 63% of its maximum value.
∴ I = I₀ [1 - e^-t/τ]

Decay of Current: If switch S is connected to B, the battery is disconnected. The current does not fall instantaneously from I₀ to zero but decays slowly due to the current induced in the coil in the direction opposite to that of the falling current. The induced e.m.f. in the inductance will be equal to -L \( \frac{dI}{dt} \) corresponding to the instantaneous current I in the resistor R at that time.
∴ -L \( \frac{dI}{dt} \) = RI

Rate of decay of current = \( \frac{dI}{dt} \) = -\( \left(\frac{R}{L}\right) \) I
or \( \frac{dI}{I} \) = -\( \left(\frac{R}{L}\right) \) dt

When t = 0, the current I₀ is maximum and the current at time t is I.
∴ ∫ \( \frac{dI}{I} \) (from I₀ to I) = -\( \frac{R}{L} \) ∫ dt (from 0 to t)
∴ I = I₀ e^-Rt/L = I₀ e^-t/τ

Where τ = \( \frac{L}{R} \) is the time constant of the circuit.

When t = \( \frac{L}{R} \), I = I₀ e^{-(R/L) × (L/R)} = \( \frac{I_0}{e} \) = \( \frac{I_0}{2.718} \) = 0.37 I₀

The time constant τ is defined as the time interval during which the current decays to 37% of the maximum current during the decay. The rate of decay of the current shows an exponential decay behaviour as shown in the figure.

[Figure: Graph of current I against time t for decay of current. The curve starts at I₀ on the I axis and falls exponentially towards the t axis; the value 0.37 I₀ is marked on the I axis with a dotted line to the curve at time τ]

The energy stored in an inductor of inductance L, when the current I is passing through it, is equal to \( \frac{1}{2} \) L I² which is in the magnetic form. Such an L-C circuit produces harmonic oscillation in an electrical circuit in which the energy changes from the electrical to magnetic and vice versa. Such oscillations can be sustained in an electrical circuit and can continue for a long time with the same amplitude if there is negligible resistance in the circuit.

Force F3 on arm CD = BIl where CD = l