Electromagnetic Induction & Lenz's Law

Electromagnetic induction is the phenomenon where a changing magnetic flux through a circuit induces an EMF (and hence a current if the circuit is closed). Discovered by Faraday, this is the principle behind generators, transformers, and most of modern electrical technology.

Faraday's Law — Bar Magnet Moving Into a Coil:

Faraday's Laws of Electromagnetic Induction:

First law: Whenever the magnetic flux through a circuit changes, an EMF is induced.
Second law: The induced EMF equals the negative rate of change of magnetic flux: $\varepsilon = -\frac{d\Phi_B}{dt}$, where $\Phi_B = \int \vec{B} \cdot d\vec{A}$.
For a coil of $N$ turns: $\varepsilon = -N\frac{d\Phi_B}{dt}$. The flux can change due to change in $B$, change in area $A$, or change in the angle between $\vec{B}$ and $\vec{A}$.

Lenz's Law: The direction of the induced current is such that it opposes the change in flux that produced it. This is a consequence of energy conservation — the induced current creates a magnetic field that opposes the flux change. Lenz's law is encoded in the negative sign of Faraday's law.

Motional EMF: When a conductor of length $l$ moves with velocity $v$ perpendicular to a uniform magnetic field $B$: $\varepsilon = Bvl$. This arises from the Lorentz force on free charges in the moving conductor.
For a rod sliding on rails: $\varepsilon = Bvl$, current $I = Bvl/R$, force on rod $F = BIl = B^2l^2v/R$, power dissipated $P = B^2l^2v^2/R$.

Rotating Coil (AC Generator): A coil of $N$ turns, area $A$, rotating with angular velocity $\omega$ in field $B$: $\Phi = NBA\cos(\omega t)$, $\varepsilon = NBA\omega\sin(\omega t)$.
Peak EMF: $\varepsilon_0 = NBA\omega$. This produces alternating current.

Eddy Currents in a Metal Plate:

Eddy Currents: When a bulk conductor moves through a magnetic field (or is exposed to changing flux), circulating currents called eddy currents are induced. They cause heating (used in induction furnaces) and braking (electromagnetic damping). Laminated cores reduce eddy current losses.

Self-Inductance — Coil With Changing Flux:

Self-Inductance: A coil opposes changes in its own current.
$\varepsilon = -L\frac{dI}{dt}$, where $L$ is the self-inductance (unit: henry, H).
For a solenoid: $L = \mu_0 n^2 Al = \mu_0 N^2 A/l$, where $l$ is length, $A$ is cross-section, $N$ is total turns.

Mutual Inductance: Two coils interact through shared flux.
EMF in coil 2 due to changing current in coil 1: $\varepsilon_2 = -M\frac{dI_1}{dt}$.
For two coaxial solenoids: $M = \mu_0 n_1 n_2 A l$ (where $A$ is the area of the inner solenoid).
$M = k\sqrt{L_1 L_2}$ where $k$ is the coupling coefficient ($0 \leq k \leq 1$).

Energy Stored in an Inductor: $U = \frac{1}{2}LI^2$. Energy density in a magnetic field: $u = B^2/(2\mu_0)$ (analogous to $u = \varepsilon_0 E^2/2$ for electric fields).