Faraday's Law: Electromagnetic Induction

Faraday's Law of induction is one of Maxwell's four equations and the foundation of electric motors, generators, transformers, and induction stoves. A changing magnetic flux through a coil induces an electromotive force (EMF) in the coil:

$\varepsilon = -N\,\dfrac{d\Phi_B}{dt}$

where $\Phi_B = B \cdot A$ is the magnetic flux through the loop and $N$ is the number of turns. The negative sign is Lenz's law: the induced current flows in whichever direction opposes the change in flux.

What Drives the EMF

It's the change in flux, not the flux itself. A magnet sitting still inside a coil produces no EMF, no matter how strong it is. But move it just a little, and current flows.

Three ways to change flux:

1. Change $B$ (e.g., move a magnet closer or farther)
2. Change $A$ (e.g., squeeze the coil)
3. Change orientation (e.g., rotate the coil — this is how an electric generator works)

Step-by-Step Solution

Given: A coil with $N = 100$ turns and area $A = 0.01\;\text{m}^{2}$. A magnet creates a field that changes from $B_1 = 0.5\;\text{T}$ to $B_2 = 0.1\;\text{T}$ in $\Delta t = 0.02\;\text{s}$.

Find: The induced EMF.

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Step 1 — Compute the initial and final flux through one turn.

$\Phi_1 = B_1\,A = (0.5)(0.01) = 0.005\;\text{Wb}$

$\Phi_2 = B_2\,A = (0.1)(0.01) = 0.001\;\text{Wb}$

Step 2 — Compute the change in flux.

$\Delta\Phi = \Phi_2 - \Phi_1 = 0.001 - 0.005 = -0.004\;\text{Wb}$

The negative sign tells us the flux decreased.

Step 3 — Find the average rate of change.

$\dfrac{d\Phi}{dt} \approx \dfrac{\Delta\Phi}{\Delta t} = \dfrac{-0.004}{0.02} = -0.2\;\text{Wb/s}$

Step 4 — Apply Faraday's Law.

$\varepsilon = -N\,\dfrac{d\Phi}{dt} = -100 \times (-0.2) = 20\;\text{V}$

Step 5 — Determine the direction of induced current (Lenz's Law).

The flux decreased, so the induced current must try to maintain the original flux — flowing in the direction that creates a magnetic field in the same direction as the original. By the right-hand rule, this determines whether the current flows clockwise or counterclockwise as viewed from the magnet's side.

Step 6 — Sanity check.

A 20 V EMF from changing flux in 0.02 s is a substantial signal. If the coil has $R = 10\;\Omega$ resistance, the induced current would be $I = V/R = 2\;\text{A}$ — definitely measurable, easily enough to power an LED.

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Answer:

$\boxed{\;\varepsilon = -N\,\dfrac{d\Phi}{dt} = 20\;\text{V}\;}$

When the field through 100 turns of a 0.01 m² coil drops from 0.5 T to 0.1 T in 20 ms, the induced EMF is 20 volts. The direction is given by Lenz's law: the induced current opposes the change in flux, in this case flowing to maintain the original (decreasing) field.

Try It

• Move the magnet position slider — see the induced current fluctuate.
• The current direction reverses when you change from inserting to withdrawing the magnet.
• Faster motion = bigger flux change rate = bigger EMF.
• Notice that holding the magnet still produces zero current, no matter how strong it is.