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.

The Formula and Its Symbols

For a coil with $N$ turns,

where $\mathcal{E}$ 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 measures how much field passes through the loop. For a flat loop in a uniform field,

where $\theta$ is the angle between the field and the loop's area vector (perpendicular to the surface). This assumes the field is uniform across the loop and the loop is flat. In that setting, flux changes in three ways: the field strength $B$ changes, the loop area $A$ changes, or the angle $\theta$ changes as the loop rotates. If none changes, the flux is constant and no emf is induced.

Why the Formula Has This Shape

Faraday's law is fundamentally a rate-of-change statement, which is why $d\Phi_B/dt$ sits at its heart. A larger change in flux over the same time gives a larger emf; the same change spread over a longer time gives a smaller one. That is why pushing a magnet quickly into a coil induces a larger emf than pushing it slowly, the flux changes faster.

The factor $N$ appears because each of the $N$ turns links the same flux, and their emfs add in series, so the coil's total emf scales with the number of turns. The minus sign and Lenz's law set the direction: the induced current creates a magnetic effect that opposes the change in flux, not necessarily the original field. If flux is increasing, the induced current acts to reduce the increase; if decreasing, it resists the decrease.

Worked Example: A Field Increases Through a Coil

A coil has $N = 50$ turns and area $A = 0.020\ \mathrm{m^2}$. A uniform field points perpendicular to the loop, so $\cos\theta = 1$, and it increases from $0.10\ \mathrm{T}$ to $0.40\ \mathrm{T}$ in $0.20\ \mathrm{s}$.

Because the field is perpendicular, the flux per turn is $\Phi_B = BA$, so the change per turn is

The average induced emf magnitude is

So the induced emf has magnitude $1.5\ \mathrm{V}$. For direction, apply Lenz's law separately: since the flux is increasing, the induced current must create a magnetic effect that opposes that increase.

Practice and Check

Keep the same coil, but let the field change happen in $0.40\ \mathrm{s}$ instead of $0.20\ \mathrm{s}$.

The flux change $\Delta\Phi_B = 0.006\ \mathrm{Wb}$ is unchanged, so doubling the time should halve the emf to $0.75\ \mathrm{V}$. As a second case, rotate the same coil instead of changing $B$: now the changing factor is $\cos\theta$, which tests the same law from a different angle. If your halved-time answer is not $0.75\ \mathrm{V}$, recheck whether you accidentally changed $\Delta\Phi_B$ as well as $\Delta t$.

Calculation Traps to Avoid

• A field is present, so there must be emf. A steady field through a steady loop induces nothing; the flux has to change.
• Using $\Phi_B = BA$ without the angle. That form is the special case $\cos\theta = 1$. In general use $\Phi_B = BA\cos\theta$.
• Reading the minus sign as just a negative number. It is mainly about direction. For a size-only question, take the magnitude and handle direction with Lenz's law.
• Forgetting the number of turns. The emf scales with $N$; dropping it makes 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 changing magnetic flux always induces emf. It is also a clean link between fields and circuits, since a changing magnetic situation creates an emf that, in a closed loop, can drive current.