Faraday's Law — Electromagnetic Induction & Lenz's Law

Faraday's law says that a changing magnetic flux through a loop induces an emf. A magnetic field by itself is not enough. If the flux through the loop stays constant, the induced emf is zero.

For a coil with $N$ turns,

$$\mathcal{E} = -N \frac{d\Phi_B}{dt}$$

Here $\mathcal{E}$ is the induced emf and $\Phi_B$ is the magnetic flux through one turn. The minus sign comes from Lenz's law: the induced current acts to oppose the change in flux.

Magnetic Flux Means Field Passing Through the Loop

Magnetic flux measures how much magnetic field passes through a loop. For a flat loop in a uniform magnetic field,

$$\Phi_B = BA \cos \theta$$

Here $\theta$ is the angle between the magnetic field and the loop's area vector, which is perpendicular to the loop's surface. This formula assumes the field is uniform across the loop and the loop can be treated as flat.

In that setting, flux can change in three standard ways:

1. The field strength $B$ changes.
2. The loop area $A$ changes.
3. The angle $\theta$ changes because the loop rotates.

If none of those changes, then the flux stays constant and no emf is induced.

Bigger Flux Change Means Bigger Emf

Faraday's law is a rate-of-change idea. A larger change in flux over the same time gives a larger emf. The same change spread over a longer time gives a smaller emf.

That is why moving a magnet quickly into a coil usually gives a larger induced emf than moving it slowly. The setup can vary, but the pattern is the same: faster flux change, larger emf.

Lenz's Law Sets the Direction

Lenz's law gives the direction of the induced effect. It says the induced current creates its own magnetic effect in a direction that opposes the change in flux.

That wording matters. The current does not always oppose the original magnetic field. It opposes the change in flux. If the flux through the loop is increasing, the induced current acts to reduce that increase. If the flux is decreasing, the induced current acts to resist the decrease.

Worked Example: A Magnetic Field Increases Through a Coil

Suppose a coil has $N = 50$ turns and area $A = 0.020\ \mathrm{m^2}$. A uniform magnetic field points perpendicular to the loop's surface, so $\cos \theta = 1$. The field increases from $0.10\ \mathrm{T}$ to $0.40\ \mathrm{T}$ in $0.20\ \mathrm{s}$.

Because the field is perpendicular to the loop, the flux through one turn is $\Phi_B = BA$. The change in flux per turn is

$$\Delta \Phi_B = A \Delta B = (0.020)(0.40 - 0.10) = 0.006\ \mathrm{Wb}$$

The average induced emf magnitude is

$$|\mathcal\{E\}| = N \frac\{|\Delta \Phi_B|\}\{\Delta t\}$$

So

$$|\mathcal\{E\}| = 50 \cdot \frac\{0.006\}\{0.20\} = 1.5\ \mathrm\{V\}$$

So the induced emf has magnitude $1.5\ \mathrm{V}$.

For direction, use Lenz's law separately. Since the magnetic flux is increasing, the induced current must create a magnetic effect that opposes that increase.

Common Mistakes

A Magnetic Field Is Present, So There Must Be Emf

A steady magnetic field through a steady loop does not induce emf. The flux has to change.

Using $\Phi_B = BA$ Without Checking the Angle

$\Phi_B = BA$ is only the special case where the field is perpendicular to the loop, so $\cos \theta = 1$. In general, use $\Phi_B = BA \cos \theta$ when its conditions apply.

Treating the Minus Sign as Only a Negative Number

The minus sign in Faraday's law is mainly about direction. If a problem asks only for the size of the emf, use the magnitude and handle direction with Lenz's law.

Forgetting the Number of Turns

For a coil, the induced emf scales with $N$. Missing that factor can make the answer much too small.

Where Faraday's Law Is Used

Faraday's law sits behind generators, transformers, induction cooktops, guitar pickups, and many sensors. The details differ, but the same core idea appears each time: changing magnetic flux induces emf.

It is also a clean link between fields and circuits. A changing magnetic situation creates an emf, and in a closed loop that emf can drive current.

Try a Similar Problem

Keep the same coil, but let the field change happen in $0.40\ \mathrm{s}$ instead of $0.20\ \mathrm{s}$. The flux change is the same, so the induced emf is cut in half.

If you want one more case, try rotating the same coil instead of changing $B$. That tests the same law from a different angle.