Chapter 8

Faraday's Law & Electromagnetic Induction

Changing magnetic flux induces EMF. The connection between E and B.

8.1 Faraday's Law of Induction

Faraday discovered experimentally that a changing magnetic flux through a circuit induces an electromotive force (EMF). In integral form:

$$\boxed{\mathcal{E} = \oint_C \mathbf{E} \cdot d\boldsymbol{\ell} = -\frac{d\Phi_B}{dt} = -\frac{d}{dt}\int_S \mathbf{B}\cdot d\mathbf{a}}$$

The differential form (applying Stokes' theorem) is:

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$

This is the third of Maxwell's equations. It tells us that a time-varying magnetic field generates a curling electric field — and vice versa (through Ampere-Maxwell).

8.1.1 Lenz's Law

The induced EMF drives a current that opposes the change in flux — this is the content of the minus sign in Faraday's law, and is known as Lenz's law. It is a consequence of energy conservation.

8.2 Inductance

The self-inductance $L$ of a circuit relates the flux through it to the current creating it: $\Phi = LI$, so $\mathcal{E} = -L\,dI/dt$.

Toroidal solenoid

$$L = \frac{\mu_0 N^2 A}{2\pi R}$$

Long solenoid

$$L = \mu_0 n^2 V = \mu_0 n^2 \pi r^2 \ell$$

The mutual inductance $M$ between two loops is given by the Neumann formula:

$$M_{12} = \frac{\mu_0}{4\pi}\oint\oint \frac{d\boldsymbol{\ell}_1 \cdot d\boldsymbol{\ell}_2}{|\mathbf{r}_1 - \mathbf{r}_2|}$$

8.2.1 Energy in Magnetic Fields

$$W = \frac{1}{2}LI^2 = \frac{1}{2\mu_0}\int B^2\,d\tau$$

The magnetic energy density is $u_B = B^2/(2\mu_0)$ — the magnetic analog of$u_E = \epsilon_0 E^2/2$.

Simulation: Faraday Induction & Mutual Inductance

Faraday Induction & Mutual Inductance

Computes EMF from changing flux, mutual inductance via the Neumann formula, and magnetic energy storage.

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ──────────────────────────────────────────────
# Faraday Induction: EMF in a loop due to changing
# magnetic flux (B increasing in time)
# Also: mutual inductance between two coaxial loops
# ──────────────────────────────────────────────
mu0 = 4 * np.pi * 1e-7

# ── 1. EMF vs time for sinusoidal B ──
B0 = 0.01    # T
omega = 2 * np.pi * 60  # 60 Hz
A = 0.01     # loop area m²
t = np.linspace(0, 0.05, 1000)
Phi = B0 * A * np.sin(omega * t)
EMF = -np.gradient(Phi, t)  # Faraday: EMF = -dΦ/dt

# ── 2. Mutual inductance Neumann formula ──
# Two coaxial circular loops, radii R1, R2, separated by d
# M = (mu0/(4pi)) * ∮∮ (dl1 · dl2)/|r1-r2|
R1, R2 = 0.1, 0.08
d_vals = np.linspace(0.01, 0.5, 80)
N_seg = 200
phi1 = np.linspace(0, 2*np.pi, N_seg, endpoint=False)
phi2 = np.linspace(0, 2*np.pi, N_seg, endpoint=False)

def mutual_inductance(R1, R2, d):
    dphi1 = phi1[1] - phi1[0]; dphi2 = phi2[1] - phi2[0]
    r1x = R1*np.cos(phi1); r1y = R1*np.sin(phi1)
    r2x = R2*np.cos(phi2); r2y = R2*np.sin(phi2)
    dl1x = -R1*np.sin(phi1)*dphi1; dl1y = R1*np.cos(phi1)*dphi1
    dl2x = -R2*np.sin(phi2)*dphi2; dl2y = R2*np.cos(phi2)*dphi2
    M = 0.0
    for i in range(N_seg):
        dx = r1x[i] - r2x; dy = r1y[i] - r2y; dz2 = d**2
        r = np.sqrt(dx**2 + dy**2 + dz2)
        r[r < 1e-10] = 1e-10
        dot = dl1x[i]*dl2x + dl1y[i]*dl2y   # dl1·dl2 (z-comps = 0)
        M += np.sum(dot / r)
    return mu0/(4*np.pi) * M

M_vals = np.array([mutual_inductance(R1, R2, d) for d in d_vals])

# ── Self-inductance of a solenoid ──
n = 1000   # turns/m
r_sol = 0.01  # radius 1 cm
L_per_m = mu0 * n**2 * np.pi * r_sol**2
print(f"Solenoid self-inductance: {L_per_m*1000:.3f} mH/m")
print(f"Mutual inductance at d=5cm: M = {mutual_inductance(R1,R2,0.05)*1e6:.3f} μH")

fig, axes = plt.subplots(1, 3, figsize=(15, 5))
fig.patch.set_facecolor('#0a0a1a')
for ax in axes:
    ax.set_facecolor('#0a0a1a')
    for sp in ax.spines.values(): sp.set_edgecolor('#334155')

axes[0].plot(t*1000, Phi*1e3, color='#60a5fa', lw=2, label=r'$Phi_B$ [mWb]')
ax0b = axes[0].twinx()
ax0b.plot(t*1000, EMF*1000, color='#f59e0b', lw=2, label='EMF [mV]')
ax0b.tick_params(colors='#94a3b8')
ax0b.set_ylabel('EMF [mV]', color='#f59e0b')
axes[0].set_xlabel('t [ms]', color='#94a3b8'); axes[0].set_ylabel(r'$Phi_B$ [mWb]', color='#60a5fa')
axes[0].tick_params(colors='#94a3b8')
axes[0].set_title(r'Faraday: EMF $= -dPhi/dt$', color='white', fontsize=11)
lines0 = [plt.Line2D([0],[0],color='#60a5fa',lw=2), plt.Line2D([0],[0],color='#f59e0b',lw=2)]
axes[0].legend(lines0, [r'$Phi_B$','EMF'], facecolor='#1e293b', labelcolor='white', fontsize=9)

axes[1].plot(d_vals*100, M_vals*1e6, color='#a78bfa', lw=2)
axes[1].set_xlabel('Separation d [cm]', color='#94a3b8'); axes[1].set_ylabel('M [μH]', color='#94a3b8')
axes[1].set_title(f'Mutual Inductance: R₁={R1*100:.0f}cm, R₂={R2*100:.0f}cm', color='white', fontsize=11)
axes[1].tick_params(colors='#94a3b8')
axes[1].grid(alpha=0.2, color='#334155')

# Energy stored in inductor
L = 1e-3  # 1 mH
I_arr = np.linspace(0, 10, 100)
W = 0.5 * L * I_arr**2
axes[2].plot(I_arr, W*1000, color='#34d399', lw=2)
axes[2].fill_between(I_arr, 0, W*1000, alpha=0.2, color='#34d399')
axes[2].set_xlabel('Current I [A]', color='#94a3b8'); axes[2].set_ylabel('Energy W [mJ]', color='#94a3b8')
axes[2].set_title(r'Magnetic Energy: $W = rac{1}{2}LI^2$, L=1mH', color='white', fontsize=11)
axes[2].tick_params(colors='#94a3b8')
axes[2].grid(alpha=0.2, color='#334155')

plt.tight_layout()
plt.savefig('faraday_induction.png', dpi=130, bbox_inches='tight', facecolor=fig.get_facecolor())

Click Run to execute the Python code

First run will download Python environment (~15MB)

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