Electromagnetism: Faraday's Law & Inductance

Faraday's Law

$\mathcal{E} = -\frac{d\Phi_B}{dt}$ where $\Phi_B = \int \vec{B} \cdot d\vec{A}$.

For $N$ loops: $\mathcal{E} = -N\frac{d\Phi_B}{dt}$.

Lenz's Law

The induced EMF opposes the change in flux. The negative sign in Faraday's law encodes Lenz's law.

Motional EMF

A conductor of length $L$ moving with velocity $v$ perpendicular to field $B$: $\mathcal{E} = BLv$

For a sliding bar on rails with resistance $R$: $I = \frac{BLv}{R}, \quad F_{\text{drag}} = \frac{B^2L^2v}{R}$

Faraday's Law in Differential Form

$\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$

A changing magnetic field creates a (non-conservative) electric field.

Self-Inductance

$\mathcal{E}_L = -L\frac{dI}{dt}$ where $L$ is the inductance. Unit: henry (H).

Solenoid inductance: $L = \mu_0 n^2 V = \mu_0 n^2 A \ell$ where $\ell$ is length and $A$ is cross-sectional area.

Energy Stored in an Inductor

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

Energy density in a magnetic field: $u = \frac{B^2}{2\mu_0}$

RL Circuits

Growth of current (RL charging): $I(t) = \frac{\mathcal{E}}{R}(1 - e^{-Rt/L})$

Time constant: $\tau = L/R$.

Decay of current: $I(t) = I_0 e^{-Rt/L}$

LC Oscillations

$Q(t) = Q_0 \cos(\omega t + \phi), \quad \omega = \frac{1}{\sqrt{LC}}$

Energy oscillates between the capacitor ($\frac{1}{2}Q^2/C$) and inductor ($\frac{1}{2}LI^2$). Total energy is conserved.

Maxwell's Equations (Overview for AP C)

1. Gauss's law (electricity): $\oint \vec{E}\cdot d\vec{A} = Q_{\text{enc}}/\epsilon_0$
2. Gauss's law (magnetism): $\oint \vec{B}\cdot d\vec{A} = 0$ (no magnetic monopoles)
3. Faraday's law: $\oint \vec{E}\cdot d\vec{l} = -d\Phi_B/dt$
4. Ampère-Maxwell: $\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}} + \mu_0\epsilon_0 d\Phi_E/dt$

Problem: A circular loop of radius $0.1\ \text{m}$ is in a magnetic field that changes as $B(t) = 0.5 + 0.3t^2$ T. Find the induced EMF at $t = 2\ \text{s}$.

Solution:

$\Phi = BA = (0.5 + 0.3t^2)\pi(0.1)^2 = 0.01\pi(0.5 + 0.3t^2)$

$\mathcal{E} = -\frac{d\Phi}{dt} = -0.01\pi(0.6t)$

At $t = 2$: $\mathcal{E} = -0.01\pi(1.2) = -0.012\pi \approx -0.0377\ \text{V}$

Magnitude: $37.7\ \text{mV}$.

• Faraday's law: $\mathcal{E} = -d\Phi_B/dt$. Lenz's law gives the direction.
• Self-inductance: $\mathcal{E} = -L\,dI/dt$; energy: $U = \frac{1}{2}LI^2$.
• RL circuits: exponential growth/decay with $\tau = L/R$.
• LC circuits: oscillations at $\omega = 1/\sqrt{LC}$.
• Maxwell's equations unify E&M.