Faraday’s Second Law of Electromagnetic Induction

1. Statement of the Law / Concept Overview

Faraday’s Second Law of electromagnetic induction states:

The magnitude of the induced emf in a closed circuit is directly proportional to the rate of change of magnetic flux through the circuit.

In simple words:

• If magnetic flux changes rapidly, a large emf is induced.
• If magnetic flux changes slowly, a small emf is induced.
• If magnetic flux does not change, no emf is induced — even if the magnetic field, area, or coil exists.

This law quantifies the amount of induced emf.

Let the magnetic flux through a coil be [\phi].
If the flux changes by an amount [\Delta \phi] in time [\Delta t], then:

[\text{Induced emf}] [\propto \dfrac{\Delta \phi}{\Delta t}]

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2. Clear Explanation and Mathematical Derivation

Consider a coil with N turns placed in a magnetic field.
Let the magnetic flux linked with one turn be [\phi].
Total flux linked with the coil:

[\Phi = N\phi]

If the flux changes with time, then:

[\text{Induced emf}] [E \propto \dfrac{d\Phi}{dt}]

Faraday introduced a proportionality constant 1, and Lenz gave the negative sign (direction of induced emf opposes change).

Thus:

[E = -\dfrac{d\Phi}{dt}]

Or for a coil with N turns:

[E = -N \dfrac{d\phi}{dt}]

This equation tells us:

• If [\dfrac{d\Phi}{dt}] is large → large emf
• If [\dfrac{d\Phi}{dt}] is small → small emf
• If [\dfrac{d\Phi}{dt} = 0] → no emf

This is the quantitative statement of electromagnetic induction.

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3. Dimensions and Units

Magnetic Flux

• Unit: Weber (Wb)
• Dimensions: [M L^{2} T^{-2} A^{-1}]

Rate of change of flux

• Unit: Weber per second (Wb s⁻¹)

Induced emf

• Unit: Volt (V)
• Dimensions: [M L^{2} T^{-3} A^{-1}]

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4. Key Features

• Induced emf depends on how fast flux changes, not on flux itself.
• If flux changes linearly with time, emf is constant.
• If flux changes sinusoidally, emf is also sinusoidal.
• More turns (N) → greater emf.
• The direction of induced emf always opposes the change (Lenz’s Law).

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5. Important Formulas to Remember

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6. Conceptual Questions with Solutions

1. Why is induced emf proportional to rate of change of flux?

Because a changing magnetic environment creates an electric field, and the faster the flux changes, the stronger the induced electric field and emf.

2. Can induced emf be zero even when magnetic field is present?

Yes. If the magnetic flux is constant (no change with time), emf is zero even with a magnetic field.

3. Why does a coil with more turns have greater induced emf?

Because total flux linkage [\Phi = N\phi] increases with N, so the induced emf becomes [E = -N \dfrac{d\phi}{dt}].

4. What happens if flux decreases instead of increasing?

Only the **sign** of emf (direction) changes; the magnitude depends only on the rate of change.

5. Why is the negative sign used in the formula?

To obey Lenz’s law — the induced emf always opposes the change in flux.

6. Does induced emf depend on area of the coil?

Indirectly, yes. If area increases, flux increases, and changing area changes flux.

7. What if flux changes instantly?

A very large emf is induced temporarily, theoretically infinite for instantaneous change.

8. Why does rotating a coil induce emf?

Rotation changes the angle θ, which changes flux [\phi = BA\cos\theta], producing emf.

9. If B is constant and A is constant, can emf still be induced?

Yes, if the orientation changes (θ changes), flux changes.

10. If flux doubles in half the time, what happens to emf?

Rate of change doubles, so induced emf doubles.

11. Can emf be induced in an open coil?

Electric field is induced, but emf across broken ends cannot drive current (no closed circuit).

12. Does induced emf depend on resistance of the coil?

No. Emf depends only on flux change. Resistance affects **current**, not **emf**.

13. Why does induced emf go to zero when flux becomes constant?

Because the derivative [\dfrac{d\Phi}{dt}] becomes zero.

14. What happens if the coil is removed from the magnetic field?

Flux decreases rapidly → emf is induced during removal.

15. Why is induced emf sometimes called “motional emf”? Because motion can cause change in flux, which induces emf.

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7. FAQ / Common Misconceptions

1. Is magnetic field necessary for emf?

Not exactly — **changing flux** is necessary. Flux may change even without a magnetic field if area/orientation changes.

2. Does increasing magnetic field always increase emf?

Only if the increase changes flux with time.

3. Is induced emf a type of potential difference?

Yes, it acts like a potential difference that can drive current.

4. Does a steady current in a solenoid induce emf?

No, because steady current produces constant flux.

5. Is induced emf always negative?

The negative sign only shows direction (opposition). Magnitude is positive.

6. Can flux change without motion?

Yes—changing magnetic field or orientation can change flux even if nothing moves.

7. Does faster rotation always induce more emf?

Yes, faster rotation → faster change in θ → greater [\dfrac{d\phi}{dt}].

8. If coil resistance increases, does emf decrease?

No, emf depends only on flux change, not resistance.

9. Can induced emf exist without induced current?

Yes, in an open circuit emf exists but no current flows.

10. Is induced emf instantaneous?

It appears the moment flux begins to change, but not before.

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8. Practice Questions (with Step-by-Step Solutions)

Q1. A flux of [0.4\ \text{Wb}] decreases uniformly to zero in 0.2 s in a coil of 20 turns. Find the induced emf.

Solution:

[\Delta\phi = 0.4\ \text{Wb}]
[\Delta t = 0.2\ \text{s}]
[N = 20]

[
E = -N \dfrac{\Delta\phi}{\Delta t}
]

[E] [= -20 \times \dfrac{0.4}{0.2}] [= -40\ \text{V}]

Magnitude: 40 V

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Q2. Flux through a coil changes from 5 mWb to 2 mWb in 10 ms. Find the induced emf in a 100-turn coil.

[\Delta\phi = 3 \times 10^{-3}\ \text{Wb}]
[\Delta t = 10^{-2}\ \text{s}]
[N = 100]

[E] [= -100 \dfrac{3 \times 10^{-3}}{10^{-2}}] [= -30\ \text{V}]

Magnitude: 30 V

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Q3. A coil with 50 turns has flux linkage [\Phi = 0.1\ \text{Wb}]. If flux becomes zero in 0.5 s, find emf.

[
E = -\dfrac{\Delta\Phi}{\Delta t}
]

[E] [= -\dfrac{0.1}{0.5}] [= -0.2\ \text{V}]

Magnitude: 0.2 V

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Q4. Flux linked with a 10-turn coil changes according to [\phi = 0.02t], where t is in seconds. Find emf.

[
E = -N \dfrac{d\phi}{dt}
]

[
\dfrac{d\phi}{dt} = 0.02
]

[E = -10 \times 0.02] [= -0.2\ \text{V}]

Magnitude: 0.2 V

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Q5. A coil experiences a change of flux linkage from 0.3 Wb to 0 Wb in 0.1 s. Calculate induced emf.

[
\Delta\Phi = 0.3\ \text{Wb}
]

[E = -\dfrac{0.3}{0.1}] [= -3\ \text{V}]

Magnitude: 3 V