Chapter 4

Conductors & Dielectrics

Electrostatic induction, capacitance, polarization, and the displacement field.

4.1 Conductors in Electrostatic Equilibrium

In a perfect conductor, charges redistribute until the electric field is zero everywhere inside. The key electrostatic properties of a conductor in equilibrium are:

$\mathbf{E} = 0$

inside the conductor

$\rho = 0$

no free charge inside; charge resides on surfaces

$V = \text{const}$

the conductor is an equipotential

$\mathbf{E} \perp \text{surface}$

field is normal to the surface just outside

$E_{\rm outside} = \sigma/\epsilon_0$

surface field proportional to surface charge density

4.1.1 Boundary Conditions

At the interface between two media, the boundary conditions follow from integrating Maxwell's equations across the interface. With surface charge $\sigma$ and surface current $\mathbf{K}$:

Normal component (from Gauss)

$$\epsilon_2 E_{2}^\perp - \epsilon_1 E_1^\perp = \sigma_f$$

Tangential component (from Faraday)

$$E_1^\parallel = E_2^\parallel$$

4.2 Capacitance

The capacitance $C$ of a conductor (or pair of conductors) is defined as the ratio of stored charge to potential:

$$C = \frac{Q}{V} \qquad \text{[Farads = C/V]}$$

Parallel plates

$$C = \frac{\epsilon_0 A}{d}$$

Spherical capacitor

$$C = 4\pi\epsilon_0\frac{ab}{b-a}$$

Cylindrical capacitor

$$C = \frac{2\pi\epsilon_0 L}{\ln(b/a)}$$

The energy stored in a capacitor is:

$$W = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$$

4.3 Dielectric Materials

In a dielectric, an applied electric field induces a polarization $\mathbf{P}$— a dipole moment per unit volume. For a linear, isotropic dielectric:

$$\mathbf{P} = \epsilon_0 \chi_e \mathbf{E}$$

where $\chi_e$ is the electric susceptibility. The displacement field $\mathbf{D}$ is:

$$\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P} = \epsilon_0(1 + \chi_e)\mathbf{E} = \epsilon_0 \epsilon_r \mathbf{E} = \epsilon \mathbf{E}$$

where $\epsilon_r = 1 + \chi_e$ is the relative permittivity (dielectric constant). Gauss's law in terms of $\mathbf{D}$:

$$\nabla \cdot \mathbf{D} = \rho_f, \qquad \oint_S \mathbf{D} \cdot d\mathbf{a} = Q_{f,\rm enc}$$

where $\rho_f$ is the free charge density (excluding bound charges in the dielectric). A parallel-plate capacitor filled with dielectric has $C = \epsilon_r \epsilon_0 A/d$.

Simulation: Capacitor with Dielectric Slab

Solves Laplace's equation by finite differences for a parallel-plate capacitor partially filled with a dielectric ($\epsilon_r = 4$), showing how the potential bends at the interface.

Parallel-Plate Capacitor with Dielectric

Finite-difference solution of Laplace's equation in a capacitor with a dielectric slab (ε_r = 4).

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

# ──────────────────────────────────────────────
# Parallel-plate capacitor with dielectric:
# solve Laplace's equation by finite differences
# ──────────────────────────────────────────────
Nx, Ny = 80, 80
V = np.zeros((Ny, Nx))
eps_r = np.ones((Ny, Nx))  # relative permittivity

# Geometry: plates at y=5 (V=+1) and y=75 (V=-1)
# Dielectric slab from y=30 to y=50 with eps_r=4
V[ 5, 10:70] = 1.0
V[74, 10:70] = -1.0
eps_r[30:50, :] = 4.0

# Fix plate potentials throughout iteration
def apply_bc(V):
    V[ 5, 10:70] =  1.0
    V[74, 10:70] = -1.0
    return V

# Gauss-Seidel iteration for Laplace's equation
for _ in range(3000):
    V_new = V.copy()
    for j in range(1, Ny-1):
        for i in range(1, Nx-1):
            if (j == 5 and 10 <= i < 70) or (j == 74 and 10 <= i < 70):
                continue
            # Account for varying permittivity
            ep = eps_r
            V_new[j,i] = (ep[j,i+1]*V[j,i+1] + ep[j,i-1]*V[j,i-1] +
                          ep[j+1,i]*V[j+1,i] + ep[j-1,i]*V[j-1,i]) /                          (ep[j,i+1] + ep[j,i-1] + ep[j+1,i] + ep[j-1,i] + 1e-10)
    V = apply_bc(V_new)

# Compute E = -grad V
Ey_num, Ex_num = np.gradient(-V)

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')

im0 = axes[0].pcolormesh(V, cmap='RdBu_r', shading='auto', vmin=-1, vmax=1)
plt.colorbar(im0, ax=axes[0], label='V [arb]')
axes[0].contour(V, levels=15, colors='white', alpha=0.5, linewidths=0.6)
# Mark dielectric slab
for ax in axes[:2]:
    ax.axhline(30, color='yellow', lw=1.2, linestyle='--', alpha=0.7, label=r'$epsilon_r=4$ slab')
    ax.axhline(50, color='yellow', lw=1.2, linestyle='--', alpha=0.7)
axes[0].set_title('Potential V (Laplace FD)', color='white', fontsize=11)
axes[0].set_xlabel('x', color='#94a3b8'); axes[0].set_ylabel('y', color='#94a3b8')
axes[0].tick_params(colors='#94a3b8')

x_idx = np.arange(Nx); y_idx = np.arange(Ny)
skip = 4
axes[1].quiver(x_idx[::skip], y_idx[::skip],
               Ex_num[::skip,::skip], Ey_num[::skip,::skip],
               color='#60a5fa', scale=20, alpha=0.8)
axes[1].contour(V, levels=12, colors='white', alpha=0.3, linewidths=0.5)
axes[1].axhline(30, color='yellow', lw=1.2, linestyle='--', alpha=0.7)
axes[1].axhline(50, color='yellow', lw=1.2, linestyle='--', alpha=0.7)
axes[1].set_title('E-field Vectors', color='white', fontsize=11)
axes[1].set_xlabel('x', color='#94a3b8'); axes[1].set_ylabel('y', color='#94a3b8')
axes[1].tick_params(colors='#94a3b8')

y_line = np.arange(Ny)
V_center = V[:, Nx//2]
axes[2].plot(y_line, V_center, color='#60a5fa', lw=2, label='V(y)')
axes[2].axvspan(30, 50, alpha=0.15, color='yellow', label=r'$epsilon_r=4$ slab')
axes[2].set_title('V along center column', color='white', fontsize=11)
axes[2].set_xlabel('y index', color='#94a3b8'); axes[2].set_ylabel('V', color='#94a3b8')
axes[2].tick_params(colors='#94a3b8')
axes[2].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)
axes[2].grid(alpha=0.2, color='#334155')

plt.tight_layout()
plt.savefig('capacitor_dielectric.png', dpi=130, bbox_inches='tight', facecolor=fig.get_facecolor())
print("Finite-difference Laplace solver converged.")
print(f"Max |V| in domain: {np.abs(V).max():.4f}")

Click Run to execute the Python code

First run will download Python environment (~15MB)

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