Chapter 1

Coulomb's Law & Electric Field

Force between charges, the electric field, superposition, and continuous distributions.

1.1 Coulomb's Law

The fundamental law governing the force between two point charges was established experimentally by Charles-Augustin de Coulomb in 1785. For charges $q_1$ and $q_2$ separated by a distance $r$, the force on $q_2$ due to $q_1$ is:

$$\boxed{\mathbf{F}_{12} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}\,\hat{r}_{12}}$$

where $\hat{r}_{12}$ points from $q_1$ to $q_2$, and $\epsilon_0 = 8.854 \times 10^{-12}\,\text{C}^2/(\text{N}\cdot\text{m}^2)$.

The constant $k = 1/(4\pi\epsilon_0) \approx 8.988 \times 10^9\,\text{N}\cdot\text{m}^2/\text{C}^2$. In vector form, with source position $\mathbf{r}'$ and field point $\mathbf{r}$:

$$\mathbf{F} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{|\mathbf{r} - \mathbf{r}'|^2}\,\hat{\mathscr{r}}, \qquad \hat{\mathscr{r}} = \frac{\mathbf{r} - \mathbf{r}'}{|\mathbf{r} - \mathbf{r}'|}$$

1.1.1 Superposition Principle

The force on a test charge $q$ due to a collection of $n$ source charges is the vector sum of the individual Coulomb forces:

$$\mathbf{F} = \frac{q}{4\pi\epsilon_0} \sum_{i=1}^{n} \frac{q_i}{|\mathbf{r} - \mathbf{r}_i|^2}\,\hat{\mathscr{r}}_i$$

Superposition is an empirical fact: electric forces do not interact with each other. This linearity is exact in classical electrodynamics and is one of the theory's most powerful features.

1.2 The Electric Field

Rather than specifying forces between every pair of charges, we introduce the concept of the electric field $\mathbf{E}$. The field due to source charges at any point $\mathbf{r}$ is defined by the force per unit positive test charge placed there:

$$\boxed{\mathbf{E}(\mathbf{r}) = \frac{\mathbf{F}}{q_{\rm test}} = \frac{1}{4\pi\epsilon_0} \sum_i \frac{q_i}{|\mathbf{r} - \mathbf{r}_i|^2}\,\hat{\mathscr{r}}_i}$$

1.2.1 Continuous Charge Distributions

For continuous distributions, the sum becomes an integral. Three cases arise depending on geometry:

Volume charge $\rho$

$$\mathbf{E} = \frac{1}{4\pi\epsilon_0}\int_{\mathcal{V}} \frac{\rho(\mathbf{r}')\,\hat{\mathscr{r}}}{{\mathscr{r}}^2}\,d\tau'$$

Surface charge $\sigma$

$$\mathbf{E} = \frac{1}{4\pi\epsilon_0}\int_{\mathcal{S}} \frac{\sigma(\mathbf{r}')\,\hat{\mathscr{r}}}{{\mathscr{r}}^2}\,da'$$

Line charge $\lambda$

$$\mathbf{E} = \frac{1}{4\pi\epsilon_0}\int_{\mathcal{L}} \frac{\lambda(\mathbf{r}')\,\hat{\mathscr{r}}}{{\mathscr{r}}^2}\,dl'$$

1.2.2 Worked Example: Infinite Line Charge

Consider an infinite wire with uniform line charge density $\lambda$. By symmetry the field points radially outward. Choosing the wire along the $z$-axis, at a perpendicular distance $s$:

The contribution from segment $dz'$ at height $z'$ contributes to the field. The $z$-components cancel by symmetry. Only the radial component survives:

$$\mathbf{E} = \frac{1}{4\pi\epsilon_0}\int_{-\infty}^{\infty} \frac{\lambda\,s\,dz'}{(s^2 + z'^2)^{3/2}}\,\hat{s}$$

Using the standard integral $\int_{-\infty}^{\infty} \frac{dz'}{(s^2+z'^2)^{3/2}} = \frac{2}{s^2}$:

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

This result falls off as $1/s$ (not $1/s^2$) because the charge extends to infinity in both directions.

Simulation: Electric Field Visualization

The code below computes and visualizes the electric field of a dipole using direct superposition of Coulomb fields on a 2D grid.

Electric Field of a Dipole

Computes the electric field by direct superposition (Coulomb's law) and plots field lines and magnitude heatmap for a +1 nC / −1 nC dipole.

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

# ──────────────────────────────────────────────
# Electric field of a discrete charge distribution
# ──────────────────────────────────────────────
k = 8.9875e9  # Coulomb constant [N·m²/C²]

# Charge positions and magnitudes (x, y, q)
charges = [
    (0.0,  0.5,  1e-9),   # +1 nC at (0, 0.5)
    (0.0, -0.5, -1e-9),   # −1 nC at (0,−0.5)  → dipole
]

# Grid
x = np.linspace(-1.5, 1.5, 40)
y = np.linspace(-1.5, 1.5, 40)
X, Y = np.meshgrid(x, y)

Ex = np.zeros_like(X)
Ey = np.zeros_like(Y)

for (cx, cy, q) in charges:
    dx = X - cx
    dy = Y - cy
    r3 = (dx**2 + dy**2)**1.5
    r3[r3 < 1e-6] = 1e-6   # avoid singularity
    Ex += k * q * dx / r3
    Ey += k * q * dy / r3

E_mag = np.sqrt(Ex**2 + Ey**2)

fig, axes = plt.subplots(1, 2, figsize=(13, 5.5))
fig.patch.set_facecolor('#0a0a1a')

for ax in axes:
    ax.set_facecolor('#0a0a1a')
    for spine in ax.spines.values():
        spine.set_edgecolor('#334155')

# ── Left: streamlines ──
axes[0].streamplot(X, Y, Ex, Ey, color=np.log10(E_mag + 1),
                   cmap='plasma', linewidth=1.2, density=1.6, arrowsize=1.2)
for (cx, cy, q) in charges:
    color = '#ef4444' if q > 0 else '#3b82f6'
    axes[0].plot(cx, cy, 'o', color=color, markersize=14, zorder=5)
    axes[0].text(cx + 0.08, cy, f'{q*1e9:+.0f} nC', color=color, fontsize=9)
axes[0].set_title('Electric Field Lines (dipole)', color='white', fontsize=13)
axes[0].set_xlabel('x [m]', color='#94a3b8')
axes[0].set_ylabel('y [m]', color='#94a3b8')
axes[0].tick_params(colors='#94a3b8')
axes[0].set_aspect('equal')

# ── Right: |E| heatmap ──
im = axes[1].pcolormesh(X, Y, np.log10(E_mag + 1), cmap='inferno', shading='auto')
plt.colorbar(im, ax=axes[1], label='log₁₀(|E| + 1)  [V/m]')
axes[1].contour(X, Y, np.log10(E_mag + 1), levels=10, colors='white', alpha=0.3, linewidths=0.6)
for (cx, cy, q) in charges:
    axes[1].plot(cx, cy, 'w+', markersize=14, markeredgewidth=2, zorder=5)
axes[1].set_title('|E| Magnitude (log scale)', color='white', fontsize=13)
axes[1].set_xlabel('x [m]', color='#94a3b8')
axes[1].set_ylabel('y [m]', color='#94a3b8')
axes[1].tick_params(colors='#94a3b8')
axes[1].set_aspect('equal')

plt.tight_layout()
plt.savefig('coulomb_field.png', dpi=130, bbox_inches='tight',
            facecolor=fig.get_facecolor())
print("Electric field of dipole: max |E| =", f"{E_mag.max():.2f} V/m")
print("Coulomb constant k =", f"{k:.4e} N·m²/C²")

Click Run to execute the Python code

First run will download Python environment (~15MB)

1.3 Electric Field Lines

Electric field lines are curves whose tangent at each point is parallel to $\mathbf{E}$. They provide a geometric picture of the field:

Lines originate on positive charges and terminate on negative charges.
The density of lines is proportional to $|\mathbf{E}|$.
Lines never cross (the field is single-valued at each point).
In a source-free region, lines neither begin nor end.

Key Properties of $\mathbf{E}$ in Electrostatics

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \qquad \text{(Gauss's law, differential form)}$$$$\nabla \times \mathbf{E} = 0 \qquad \text{(electrostatic field is conservative)}$$

The curl-free condition $\nabla \times \mathbf{E} = 0$ means we can write $\mathbf{E} = -\nabla V$for a scalar potential $V$.

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