Electromagnetic Induction and Faraday's Law

Faraday’s discovery powers civilization: move a magnet near a coil and current flows — mechanical motion transmuted into electricity. The rule has a stubborn personality (Lenz’s law): induced currents always fight the change that made them, which is why generators resist being turned and why that resistance is the coal you burn.

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: the induced current opposes the change in flux.

Ways to change flux

• Change $B$ (strengthen/weaken the field).
• Change $A$ (expand/shrink the loop).
• Change $\theta$ (rotate the loop).

Motional EMF — A conductor of length $L$ moving with velocity $v$ through field $B$:

$$\varepsilon = BLv$$

Self-inductance — A coil opposes changes in its own current:

$$\varepsilon = -L\frac{dI}{dt}, \qquad L = \frac{N\Phi_B}{I}$$

Energy stored in an inductor:

$$U = \frac{1}{2}LI^2$$

Key insight: Faraday's law is the basis of generators, transformers, and wireless charging. A changing magnetic field creates an electric field — even in empty space.

Common pitfall: Induction responds to changing flux, not to flux itself. A coil soaked in an enormous steady field induces nothing; a tiny field that wobbles induces plenty. Check $d\Phi/dt$, never $\Phi$.

Faraday's Law of Induction

A changing magnetic flux through a loop induces an EMF:

$$\mathcal{E} = -\frac{d\Phi_B}{dt}$$

where $\Phi_B = \int \vec{B} \cdot d\vec{A}$ is the magnetic flux. The negative sign is Lenz's law: the induced current opposes the change that caused it.

For a coil of $N$ turns: $\mathcal{E} = -N\,d\Phi_B/dt$.

Faraday's law is the principle behind generators, transformers, and wireless charging.