Electromagnetic Induction

Notes

Faraday's and Lenz's laws

• Magnetic flux: $\displaystyle \Phi_B = \int\vec B \cdot d\vec A$.
• Faraday's law: $\displaystyle \varepsilon = -N\frac{d\Phi_B}{dt}$. The flux can change because $B$ changes, the area changes, or the angle changes.
• Lenz's law: the induced current makes a field that opposes the change in flux (the minus sign).
• Motional emf: a rod of length $\ell$ moving at $v$ through $B$: $\varepsilon = B\ell v$. If it's part of a circuit, the induced current feels a force that opposes the motion (magnetic braking).
• Induced electric fields: $\displaystyle \oint\vec E \cdot d\vec\ell = -\frac{d\Phi_B}{dt}$.

Inductance

• An inductor resists changes in current: $\displaystyle \varepsilon_L = -L\frac{dI}{dt}$.
• Solenoid inductance: $L = \mu_0n^2A\ell$. Energy stored: $\displaystyle U = \frac{1}{2}LI^2$.

LR circuits

• Current growing after a switch closes: $\displaystyle I(t) = \frac{\varepsilon}{R}\left(1 - e^{-t/\tau}\right)$, with $\displaystyle \tau = \frac{L}{R}$.
• Right after the switch closes, an inductor acts like a break (no current through it). After a long time, it acts like a wire. That's the opposite of a capacitor.

LC circuits

Energy sloshes between the capacitor and the inductor, like a mass on a spring. $\displaystyle \omega = \frac{1}{\sqrt{LC}}$.

Maxwell's equations

Gauss's law, Gauss's law for magnetism ($\oint\vec B \cdot d\vec A = 0$), Faraday's law, and the Ampère–Maxwell law together describe all of electromagnetism and predict electromagnetic waves.