CBSE Class 12 Physics Electromagnetic Induction Notes Set 03

1. Experiments of Faraday and Henry

Electromagnetic induction is the production of an emf (and a current, if the circuit is closed) in a conductor or coil when the magnetic field linked with it changes. It was discovered in 1830–31 by Michael Faraday in England and Joseph Henry in the USA through a set of simple experiments using a coil connected to a galvanometer.

(i) Magnet and coil: When the north pole of a bar magnet is pushed towards a coil, the galvanometer deflects. When the magnet is pulled away, it deflects the other way. There is no deflection while the magnet is held still. A faster movement gives a larger deflection, and the same result is obtained by moving the coil towards a fixed magnet; only relative motion matters.
(ii) Coil and coil: If the bar magnet is replaced by a second coil carrying a steady current from a battery, moving either coil relative to the other again deflects the galvanometer.
(iii) Switching a current: Two coils are kept at rest close to each other. When the key in the circuit of the second coil is pressed, the galvanometer in the first coil gives a momentary deflection; when the key is released, it deflects momentarily the other way. With a steady current there is no deflection. Placing an iron rod along the common axis makes the deflection much larger.

Conclusion: An emf is induced in a coil whenever the magnetic flux linked with it changes, whatever the cause of that change. It lasts only as long as the flux keeps changing.

2. Magnetic Flux

The magnetic flux through a plane area A placed in a uniform magnetic field B is
Φ_B = \( \vec{B} \cdot \vec{A} \) = BA cos θ
where θ is the angle between B and the normal to the area. For a non-uniform field or a curved surface, Φ_B = \( \int \vec{B} \cdot d\vec{A} \).
1. Magnetic flux is a scalar quantity.
2. SI unit: weber (Wb). 1 Wb = 1 T m² = 1 V s. Dimensions: [M L² T⁻² A⁻¹].
3. Flux is maximum (BA) when the plane of the coil is perpendicular to B (θ = 0°) and zero when the plane is parallel to B (θ = 90°).
4. The flux through a coil can be changed by changing B, by changing the area A inside the field, or by changing the angle θ (rotating the coil).

3. Faraday's Law of Induction

The magnitude of the induced emf in a circuit is equal to the time rate of change of the magnetic flux through the circuit.
ε = \( -\frac{d\Phi_B}{dt} \)
For a closely wound coil of N turns, each linked with the same flux,
ε = \( -N \frac{d\Phi_B}{dt} \)
1. The product NΦ_B is called the flux linkage.
2. If the flux changes from Φ₁ to Φ₂ in time Δt, the average induced emf is ε = \( -N \frac{\Phi_2 - \Phi_1}{\Delta t} \).
3. The emf depends on how fast the flux changes, not on the value of the flux itself. A large but steady flux induces no emf.
4. In a closed circuit of resistance R, the induced current is I = ε/R, and the total charge that flows is q = \( \frac{N\,\Delta\Phi_B}{R} \), independent of how quickly the change happens.
5. The negative sign shows the direction of the induced emf, given by Lenz's law.

4. Lenz's Law and Conservation of Energy

The polarity of the induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it.

(i) N pole approaching a coil: The flux through the coil increases. The induced current flows anticlockwise as seen from the magnet, so the near face of the coil becomes a north pole and repels the approaching magnet.
(ii) N pole moving away: The flux decreases. The current flows clockwise as seen from the magnet, the near face becomes a south pole, and it attracts the receding magnet, again opposing the motion.
(iii) Increasing current in a nearby coaxial coil: The induced current in the second coil flows opposite to the current in the first coil; for a decreasing current it flows in the same direction.

Lenz's law and conservation of energy: Suppose the induced current helped the change instead of opposing it. Then a magnet given a small push towards a coil would be pulled in faster and faster, gaining kinetic energy and producing electrical energy with no work done by anyone. This is impossible because it violates the law of conservation of energy. Because the induced current opposes the motion, a person moving the magnet must do work against this opposing force, and it is this mechanical work that appears as electrical energy (and finally as heat) in the coil. So Lenz's law is a consequence of the conservation of energy.

5. Motional Electromotive Force

An emf induced in a conductor because it moves through a magnetic field is called motional emf.

Consider a straight rod PQ of length l sliding with velocity v on two parallel conducting rails, in a uniform magnetic field B perpendicular to the plane of the rails. The rails are joined by a resistor R. When PQ is at distance x from the resistor, the area of the circuit is lx.
Φ_B = Blx
ε = \( -\frac{d\Phi_B}{dt} \) = \( -Bl \frac{dx}{dt} \), and since \( \frac{dx}{dt} \) = v, the magnitude of the emf is
ε = Blv

Explanation using the Lorentz force: The free charges in the rod move with velocity v along with it, so each charge q experiences a magnetic force qvB along the rod. The work done in moving the charge from one end of the rod to the other is qvBl, so the work done per unit charge, which is the emf, is ε = vBl. This explanation applies to a conductor moving in a steady field; when a stationary coil is in a changing field, the emf is produced by an induced electric field instead.
1. The emf is zero if the rod moves parallel to B or along its own length.
2. If v makes an angle θ with B (rod still perpendicular to both), ε = Blv sin θ.
3. Rotating rod: A rod of length l rotating with angular speed ω about one end, in a plane perpendicular to a uniform field B, has an emf between its ends of ε = \( \frac{1}{2} B \omega l^2 \).
4. The direction of the induced current can be found from Lenz's law, or from Fleming's right-hand rule (thumb along motion, forefinger along B, middle finger gives the induced current).

6. Inductance

For a coil (or a pair of coils) in which no magnetic material changes, the flux linkage is directly proportional to the current producing it. The constant of proportionality is called inductance.
1. SI unit: henry (H). 1 H = 1 Wb A⁻¹ = 1 V s A⁻¹ = 1 Ω s.
2. Dimensions: [M L² T⁻² A⁻²].
3. Inductance is a scalar. It depends only on the geometry of the coils (number of turns, area, length, separation and orientation) and on the permeability of the medium, not on the current.

7. Mutual Inductance

When a changing current in one coil induces an emf in a nearby coil, the effect is called mutual induction. If a current I₂ in coil 2 produces a flux linkage N₁Φ₁ in coil 1,
N₁Φ₁ = M₁₂ I₂
where M₁₂ is the mutual inductance of coil 1 with respect to coil 2.

Two long coaxial solenoids: Let the inner solenoid S₁ have radius r₁ and n₁ turns per unit length, and the outer solenoid S₂ have n₂ turns per unit length, both of length l. A current I₂ in S₂ produces a field B₂ = μ₀n₂I₂ inside it, and the flux linkage with S₁ is (n₁l)(πr₁²)(μ₀n₂I₂). Hence
M₁₂ = μ₀ n₁ n₂ π r₁² l
Working the other way round (current in S₁, flux in S₂) gives the same result, so M₁₂ = M₂₁ = M.
1. If the space inside is filled with a material of relative permeability μ_r, M = μ_rμ₀ n₁ n₂ π r₁² l. A soft-iron core therefore greatly increases M.
2. The emf induced in coil 1 by a changing current in coil 2 is ε₁ = \( -M \frac{dI_2}{dt} \).
3. A steady current in one coil induces no emf in the other.

8. Self-Inductance

When the current in a coil changes, the flux linked with the coil itself changes and an emf is induced in the same coil. This is called self-induction.
NΦ_B = LI and ε = \( -L \frac{dI}{dt} \)
where L is the self-inductance (coefficient of self-induction) of the coil.
1. The self-induced emf opposes any change of current in the coil, so it is called a back emf.
2. Self-inductance plays the same role in a circuit that mass plays in mechanics: it resists changes in current. It is therefore called the electrical inertia of the circuit.
3. Long solenoid of length l, cross-sectional area A, n turns per unit length (N = nl turns in all): B = μ₀nI, and the flux linkage is (nl)(μ₀nI)A, so
L = μ₀ n² A l = \( \frac{\mu_0 N^2 A}{l} \)
4. With a core of relative permeability μ_r, L = μ_rμ₀ n² A l.
5. For a given length and area, L ∝ N²: doubling the number of turns makes the inductance four times.

9. Energy Stored in an Inductor

To build up a current in an inductor, work must be done against the back emf. When the current is I and increasing, the rate of doing work is
\( \frac{dW}{dt} \) = |ε| I = LI \( \frac{dI}{dt} \), so W = \( \int_0^I L I \, dI \) = \( \frac{1}{2} L I^2 \)
This energy is stored in the magnetic field of the inductor:
U = \( \frac{1}{2} L I^2 \)
It is the magnetic counterpart of the kinetic energy \( \frac{1}{2} m v^2 \) of a moving body, with L in place of m and I in place of v.

10. AC Generator

Principle: An AC generator converts mechanical energy into electrical energy using electromagnetic induction. A coil rotated in a magnetic field has a continuously changing flux through it, so an alternating emf is induced in it.
Main parts: (i) the armature, a coil of N turns wound on a core; (ii) a strong magnetic field, from permanent magnets or electromagnets; (iii) two slip rings fixed to the ends of the coil, which rotate with it; (iv) two carbon brushes pressing against the slip rings, which connect the coil to the external circuit.

Working: The coil (area A) rotates with constant angular speed ω in a uniform field B. If the normal to the coil is along B at t = 0, the angle at time t is θ = ωt and
Φ_B = BA cos ωt
ε = \( -N \frac{d\Phi_B}{dt} \) = NBAω sin ωt
ε = ε₀ sin ωt, where ε₀ = NBAω = NBA(2πν) is the peak emf.

1. The emf is maximum (ε = ±ε₀) when the plane of the coil is parallel to B: the flux is then zero but changing fastest.
2. The emf is zero when the plane of the coil is perpendicular to B: the flux is then maximum but momentarily not changing.
3. The emf reverses its direction every half rotation, so the current in the external circuit is alternating, with frequency ν = ω/2π equal to the number of rotations per second. The mains supply in India has a frequency of 50 Hz.
4. In commercial generators the mechanical energy comes from falling water (hydroelectric), from steam turbines driven by burning coal or by nuclear fuel, or from wind.

11. Quick Formula Recap