Faraday's Law & Induced EMF Worked Examples Cheat Sheet

A printable reference covering magnetic flux, Faraday’s law, Lenz’s law, induced EMF cases, and rotating coil generators for grades 11-12.

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Faraday’s law explains how changing magnetic flux creates an induced electromotive force, or EMF, in a wire loop or coil. This cheat sheet helps students connect flux changes, coil motion, magnetic field strength, and circuit orientation. It is especially useful for worked examples involving changing area, changing field, and rotating generators. Students need these tools to solve induction problems clearly and choose the correct sign and direction.

Key Facts

Magnetic flux through a flat loop is $\Phi_B = BA\cos\theta$ , where $\theta$ is the angle between the magnetic field and the area vector.
Faraday’s law for one loop is $\mathcal{E} = -\frac{\Delta \Phi_B}{\Delta t}$ , and for $N$ identical turns it is $\mathcal{E} = -N\frac{\Delta \Phi_B}{\Delta t}$ .
Lenz’s law says the induced current flows in the direction that creates a magnetic field opposing the change in magnetic flux.
If the magnetic field changes while area and angle stay constant, the induced EMF magnitude is $|\mathcal{E}| = NA\left|\frac{\Delta B}{\Delta t}\right|$ .
If the loop area changes while field and angle stay constant, the induced EMF magnitude is $|\mathcal{E}| = NB\left|\frac{\Delta A}{\Delta t}\right|$ .
For a straight conductor of length $L$ moving at speed $v$ perpendicular to a magnetic field, the motional EMF is $\mathcal{E} = BLv$ .
For a rotating coil generator with $N$ turns, area $A$ , magnetic field $B$ , and angular speed $\omega$ , the flux is $\Phi_B = BA\cos(\omega t)$ and the induced EMF is $\mathcal{E} = NBA\omega\sin(\omega t)$ .
The maximum generator EMF is $\mathcal{E}_{\max} = NBA\omega$ , which increases with more turns, stronger magnetic field, larger area, or faster rotation.

Vocabulary

Magnetic flux: Magnetic flux is the amount of magnetic field passing through a surface, calculated by $\Phi_B = BA\cos\theta$ for a uniform field.
Induced EMF: Induced EMF is the voltage produced when the magnetic flux through a circuit changes.
Faraday’s law: Faraday’s law states that the induced EMF equals the negative rate of change of magnetic flux, $\mathcal{E} = -N\frac{\Delta \Phi_B}{\Delta t}$ .
Lenz’s law: Lenz’s law states that induced current opposes the change in magnetic flux that caused it.
Area vector: The area vector is a vector perpendicular to a loop’s surface, used to measure the angle $\theta$ in $\Phi_B = BA\cos\theta$ .
Generator: A generator converts mechanical rotation into electrical energy by using changing magnetic flux to induce EMF.

Common Mistakes to Avoid

Using $\sin\theta$ instead of $\cos\theta$ for flux is wrong because $\theta$ is measured between the magnetic field and the area vector, so $\Phi_B = BA\cos\theta$ .
Forgetting the factor $N$ gives an EMF that is too small because each turn of the coil contributes to the total induced EMF, so $\mathcal{E} = -N\frac{\Delta \Phi_B}{\Delta t}$ .
Treating the negative sign as a negative voltage only is wrong because the minus sign in Faraday’s law represents Lenz’s law and the opposition to flux change.
Using the final flux instead of the change in flux is wrong because induction depends on $\Delta \Phi_B$ , not just $\Phi_B$ at one instant.
Ignoring units can hide errors because flux is measured in webers, where $1\ \text{Wb} = 1\ \text{T}\cdot\text{m}^2$ , and EMF is measured in volts.

Practice Questions

1 A coil has $N = 50$ turns and area $A = 0.020\ \text{m}^2$ . A perpendicular magnetic field changes from $B_i = 0.10\ \text{T}$ to $B_f = 0.60\ \text{T}$ in $\Delta t = 0.25\ \text{s}$ . Find the magnitude of the induced EMF.
2 A metal rod of length $L = 0.40\ \text{m}$ moves at $v = 5.0\ \text{m/s}$ perpendicular to a magnetic field of $B = 0.30\ \text{T}$ . Calculate the motional EMF $\mathcal{E}$ .
3 A generator coil has $N = 200$ , $A = 0.015\ \text{m}^2$ , $B = 0.50\ \text{T}$ , and angular speed $\omega = 120\ \text{rad/s}$ . Find $\mathcal{E}_{\max}$ .
4 A loop is pulled out of a region where the magnetic field points into the page. Explain, using Lenz’s law, whether the induced current is clockwise or counterclockwise.

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