Magnetic Flux and Lenz's Law Worked Examples Cheat Sheet

Key Facts

• Magnetic flux through a flat loop is $\Phi_B = BA\cos\theta$, where $B$ is magnetic field strength, $A$ is area, and $\theta$ is measured from the area vector.
• The SI unit of magnetic flux is the weber, with $1\,\text{Wb} = 1\,\text{T}\cdot\text{m}^2$.
• Faraday's law for a coil with $N$ turns is $\mathcal{E} = -N\frac{\Delta \Phi_B}{\Delta t}$ for average induced emf.
• For changing field strength with constant area and angle, the induced emf magnitude is $|\mathcal{E}| = N A\left|\frac{\Delta B}{\Delta t}\right|\cos\theta$.
• For a rotating loop, the flux can be written as $\Phi_B = BA\cos\omega t$ when the area vector rotates with angular speed $\omega$.
• Lenz's law says the induced current flows in the direction that creates a magnetic field opposing the change in magnetic flux.
• If magnetic flux into the page increases, the induced current produces a field out of the page, so the current is counterclockwise.
• If the loop is part of a circuit with resistance $R$, the induced current magnitude is $I = \frac{|\mathcal{E}|}{R}$.

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.

Area Vector

The area vector is a vector perpendicular to a surface with magnitude equal to the surface area.

Induced EMF

Induced emf is the voltage produced when magnetic flux through a loop changes over time.

Faraday's Law

Faraday's law states that the induced emf in a coil is $\mathcal{E} = -N\frac{\Delta \Phi_B}{\Delta t}$.

Lenz's Law

Lenz's law states that an induced current creates a magnetic field that opposes the change in flux that produced it.

Right-Hand Rule

The right-hand rule relates current direction to magnetic field direction by curling the fingers with current so the thumb points along the loop's field.