Electromagnetic Induction and Faraday's Law

Faraday's Law & Flux

Move a magnet near a coil and current flows: mechanical motion transmuted into electricity. A changing magnetic flux through a loop induces an EMF (voltage):

$$\varepsilon = -\frac{d\Phi_B}{dt}, \qquad \Phi_B = \int_S \vec{B}\cdot d\vec{A}$$

The minus sign is Lenz's law: induced currents always fight the change that made them.

Ways to change flux: Change $B$ (field strength), change $A$ (loop area), or change $\theta$ (loop orientation).

EMF, Inductance & Pitfalls

Motional EMF occurs when a conductor of length $L$ moves at velocity $v$ through a field $B$:

$$\varepsilon = BLv$$

Self-inductance ($L$) measures how a coil opposes changes in its own current:

$$\varepsilon = -L\frac{dI}{dt}, \qquad U = \frac{1}{2}LI^2$$

where $U$ is the energy stored in an inductor and $L = N\Phi_B / I$.

Common pitfall: Induction responds to changing flux, not to flux itself. A coil soaked in a steady field induces zero voltage; only a wobbling field creates an EMF.