Chapter 2

Gauss's Law & Applications

Electric flux, Gauss's law in integral and differential form, and symmetric applications.

2.1 Electric Flux

The electric flux through a surface element $d\mathbf{a} = da\,\hat{n}$ is:

$$d\Phi_E = \mathbf{E} \cdot d\mathbf{a}$$

For a closed surface the total flux is $\Phi_E = \oint_S \mathbf{E} \cdot d\mathbf{a}$. The key insight: for a point charge $q$ enclosed by any closed surface, the flux depends only on $q$, not on the size or shape of the surface. This is Gauss's law.

2.2 Gauss's Law

Integral Form

$$\boxed{\oint_S \mathbf{E} \cdot d\mathbf{a} = \frac{Q_{\rm enc}}{\epsilon_0}}$$

The total electric flux through any closed surface equals the total charge enclosed divided by $\epsilon_0$.

2.2.1 Derivation from Coulomb's Law

Start with the field of a single point charge at the origin. For a spherical Gaussian surface of radius $r$:

$$\oint \mathbf{E} \cdot d\mathbf{a} = E(r) \cdot 4\pi r^2 = \frac{q}{4\pi\epsilon_0 r^2} \cdot 4\pi r^2 = \frac{q}{\epsilon_0}$$

For a non-spherical surface: the solid angle argument shows that $\mathbf{E} \cdot d\mathbf{a} = (q/4\pi\epsilon_0)(d\Omega)$where $d\Omega$ is the solid angle subtended. Integrating over the closed surface gives$4\pi$ steradians if $q$ is inside, and zero if outside. By superposition, Gauss's law holds for any charge distribution.

2.2.2 Differential Form — Divergence Theorem

Applying the divergence theorem $\oint_S \mathbf{E} \cdot d\mathbf{a} = \int_\mathcal{V} \nabla \cdot \mathbf{E}\,d\tau$and writing $Q_{\rm enc} = \int_\mathcal{V} \rho\,d\tau$, since the volume is arbitrary:

$$\boxed{\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}} \qquad \text{(Gauss's law, differential form)}$$

This is the first of Maxwell's four equations. It states that the divergence of $\mathbf{E}$is proportional to the local charge density $\rho$.

2.3 Symmetric Applications

Gauss's law is most powerful when the charge distribution has spherical, cylindrical, or planar symmetry, reducing the surface integral to simple algebra.

Spherical symmetry: uniformly charged sphere (radius $R$)

$$E(r) = \begin{cases} \dfrac{1}{4\pi\epsilon_0}\dfrac{Qr}{R^3} & r < R \\[10pt] \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{r^2} & r > R \end{cases}$$

Inside, the enclosed charge grows as $r^3$; outside, the sphere looks like a point charge.

Cylindrical symmetry: infinite line charge density $\lambda$

$$\mathbf{E} = \frac{\lambda}{2\pi\epsilon_0 s}\,\hat{s}$$

Gaussian surface: cylinder of radius $s$ and length $L$; end caps contribute zero flux.

Planar symmetry: infinite sheet with surface charge $\sigma$

$$\mathbf{E} = \frac{\sigma}{2\epsilon_0}\,\hat{n}$$

Gaussian surface: pillbox straddling the sheet; the field is uniform and perpendicular to the sheet. Between two parallel plates of opposite charge: $E = \sigma/\epsilon_0$.

Numerical Verification of Gauss's Law

The simulation numerically integrates $\oint \mathbf{E} \cdot d\mathbf{a}$ over a spherical Gaussian surface and compares with $Q_{\rm enc}/\epsilon_0$, confirming Gauss's law to sub-percent accuracy.

Gauss's Law: Numerical Verification

Numerically integrates E·dA over a Gaussian sphere and verifies it equals Q_enc/ε₀ to high precision.

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

# ──────────────────────────────────────────────
# Verify Gauss's Law numerically by integrating
# E·dA over a Gaussian surface (sphere) and
# comparing with Q_enc / ε₀
# ──────────────────────────────────────────────
eps0 = 8.854e-12
k = 1.0 / (4 * np.pi * eps0)

# Point charges inside/outside
charges_inside = [(0.0, 0.0, 0.0, 1e-9)]   # 1 nC at origin
charges_outside = [(2.0, 0.0, 0.0, 2e-9)]  # 2 nC outside sphere

all_charges = charges_inside + charges_outside
R_sphere = 1.0   # Gaussian surface radius

# Discretize sphere surface
N_theta, N_phi = 80, 160
theta = np.linspace(0, np.pi, N_theta)
phi   = np.linspace(0, 2*np.pi, N_phi)
TH, PH = np.meshgrid(theta, phi)

# Surface points
sx = R_sphere * np.sin(TH) * np.cos(PH)
sy = R_sphere * np.sin(TH) * np.sin(PH)
sz = R_sphere * np.cos(TH)

# E field at each surface point
Ex = np.zeros_like(sx); Ey = np.zeros_like(sy); Ez = np.zeros_like(sz)
for (cx, cy, cz, q) in all_charges:
    dx = sx - cx; dy = sy - cy; dz = sz - cz
    r3 = (dx**2 + dy**2 + dz**2)**1.5
    r3[r3 < 1e-10] = 1e-10
    Ex += k * q * dx / r3
    Ey += k * q * dy / r3
    Ez += k * q * dz / r3

# Outward normal = r_hat on a sphere
En = (Ex * sx + Ey * sy + Ez * sz) / R_sphere

# Area element dA = R²sinθ dθ dφ
dtheta = np.pi / N_theta
dphi = 2 * np.pi / N_phi
dA = R_sphere**2 * np.abs(np.sin(TH)) * dtheta * dphi

flux_numerical = np.sum(En * dA)
Q_enc = sum(q for (_, _, _, q) in charges_inside)
flux_theory = Q_enc / eps0

print(f"Numerical flux ∮E·dA  = {flux_numerical:.4e} V·m")
print(f"Theory Q_enc/ε₀       = {flux_theory:.4e} V·m")
print(f"Relative error        = {abs(flux_numerical - flux_theory)/abs(flux_theory)*100:.3f}%")

# ── Visualization: E-field of point charge + Gaussian sphere ──
fig, axes = plt.subplots(1, 2, figsize=(12, 5))
fig.patch.set_facecolor('#0a0a1a')
for ax in axes:
    ax.set_facecolor('#0a0a1a')
    for sp in ax.spines.values():
        sp.set_edgecolor('#334155')

# E-field cross-section (z=0 plane)
x1d = np.linspace(-2.5, 2.5, 60)
Xg, Yg = np.meshgrid(x1d, x1d)
EGx = np.zeros_like(Xg); EGy = np.zeros_like(Yg)
for (cx, cy, cz, q) in all_charges:
    dx = Xg - cx; dy = Yg - cy
    r3 = (dx**2 + dy**2 + 0.01)**1.5
    EGx += k * q * dx / r3
    EGy += k * q * dy / r3

E_m = np.sqrt(EGx**2 + EGy**2)
axes[0].streamplot(Xg, Yg, EGx, EGy, color=np.log10(E_m + 1),
                   cmap='plasma', density=1.8, linewidth=1.0, arrowsize=1.2)
circle = plt.Circle((0, 0), R_sphere, color='cyan', fill=False, linewidth=2, linestyle='--', label='Gaussian surface')
axes[0].add_patch(circle)
for (cx, cy, cz, q) in all_charges:
    c = '#ef4444' if q > 0 else '#3b82f6'
    axes[0].plot(cx, cy, 'o', color=c, markersize=12, zorder=5)
axes[0].set_xlim(-2.5, 2.5); axes[0].set_ylim(-2.5, 2.5)
axes[0].set_aspect('equal')
axes[0].set_title('Field Lines & Gaussian Surface', color='white', fontsize=12)
axes[0].set_xlabel('x [m]', color='#94a3b8'); axes[0].set_ylabel('y [m]', color='#94a3b8')
axes[0].tick_params(colors='#94a3b8')
axes[0].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)

# Radial E vs r for spherical shell
r_vals = np.linspace(0.1, 3.0, 300)
q_inner = charges_inside[0][3]
E_radial = k * q_inner / r_vals**2
axes[1].plot(r_vals, E_radial, color='#60a5fa', linewidth=2, label=r'$E_r = kq/r^2$')
axes[1].axvline(R_sphere, color='cyan', linestyle='--', linewidth=1.5, label='Gaussian surface')
axes[1].fill_between(r_vals[r_vals < R_sphere], 0, E_radial[r_vals < R_sphere],
                     alpha=0.15, color='#ef4444', label='Inside sphere')
axes[1].fill_between(r_vals[r_vals >= R_sphere], 0, E_radial[r_vals >= R_sphere],
                     alpha=0.15, color='#3b82f6', label='Outside sphere')
axes[1].set_title('Radial E-field: Point Charge', color='white', fontsize=12)
axes[1].set_xlabel('r [m]', color='#94a3b8'); axes[1].set_ylabel('E [V/m]', color='#94a3b8')
axes[1].tick_params(colors='#94a3b8')
axes[1].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)
axes[1].grid(alpha=0.2, color='#334155')

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

Click Run to execute the Python code

First run will download Python environment (~15MB)

← Coulomb's Law Electric Potential →