Chapter 15

Retarded Potentials & Liénard–Wiechert Fields

Fields of moving charges include retardation — information travels at speed c.

15.1 Retarded Potentials

The solution to the Lorenz-gauge wave equations for time-varying sources is the retarded potentials — signals travel at speed $c$, so the potential at $(\mathbf{r}, t)$ is determined by the source at the retarded time $t_r = t - |\mathbf{r} - \mathbf{r}'|/c$:

$$V(\mathbf{r}, t) = \frac{1}{4\pi\epsilon_0}\int \frac{\rho(\mathbf{r}', t_r)}{|\mathbf{r}-\mathbf{r}'|}\,d\tau'$$$$\mathbf{A}(\mathbf{r}, t) = \frac{\mu_0}{4\pi}\int \frac{\mathbf{J}(\mathbf{r}', t_r)}{|\mathbf{r}-\mathbf{r}'|}\,d\tau'$$

These are the Jefimenko equations when expressed directly in terms of $\mathbf{E}$ and $\mathbf{B}$.

15.2 Liénard–Wiechert Potentials

For a point charge $q$ moving along trajectory $\mathbf{w}(t)$ with velocity$\mathbf{v} = \dot{\mathbf{w}}$, the retarded potentials are the Liénard–Wiechert potentials:

$$V(\mathbf{r}, t) = \frac{1}{4\pi\epsilon_0}\frac{qc}{\mathscr{r}c - \boldsymbol{\mathscr{r}}\cdot\mathbf{v}}, \qquad \mathbf{A}(\mathbf{r},t) = \frac{\mathbf{v}}{c^2}V$$

where $\boldsymbol{\mathscr{r}} = \mathbf{r} - \mathbf{w}(t_r)$ is evaluated at the retarded time. The resulting fields contain two parts:

Velocity (Coulomb) fields

Fall off as $1/\mathscr{r}^2$. Dominate near the charge. Do not radiate (no net energy flux to infinity).

Acceleration (radiation) fields

Fall off as $1/\mathscr{r}$. Proportional to acceleration. These radiate energy to infinity.

$$\mathbf{E} = \frac{q}{4\pi\epsilon_0}\frac{\mathscr{r}}{\left(\boldsymbol{\mathscr{r}}\cdot\mathbf{u}\right)^3}\left[(c^2 - v^2)\mathbf{u} + \boldsymbol{\mathscr{r}}\times(\mathbf{u}\times\mathbf{a})\right], \quad \mathbf{u} \equiv c\hat{\mathscr{r}} - \mathbf{v}$$

Simulation: Dipole Radiation & Retarded Fields

Visualizes the radiation pattern of an oscillating electric dipole, the retardation delay, and the $P \propto \omega^4$ (Larmor) frequency dependence.

Liénard–Wiechert: Dipole Radiation

Far-field radiation pattern of an oscillating electric dipole, retarded fields, and Larmor power vs frequency.

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

# ──────────────────────────────────────────────
# Liénard-Wiechert fields: radiation from an
# oscillating charge (electric dipole)
# q oscillates along z: z(t) = d*cos(omega*t)
# ──────────────────────────────────────────────
c = 3e8
mu0 = 4*np.pi*1e-7
q = 1e-9        # 1 nC
d = 0.1         # 10 cm amplitude
f = 1e8         # 100 MHz
omega = 2*np.pi*f
lam = c/f

print(f"Frequency: f = {f/1e6:.0f} MHz")
print(f"Wavelength: λ = {lam:.2f} m")
print(f"d/λ = {d/lam:.4f}  (short dipole: d << λ)")

# Radiation pattern of electric dipole
theta = np.linspace(0, 2*np.pi, 500)
# |E_theta|^2 ~ sin^2(theta) for dipole radiation
S_dipole = np.sin(theta)**2

fig, axes = plt.subplots(2, 2, figsize=(13, 10), subplot_kw=None)
fig.patch.set_facecolor('#0a0a1a')

# Polar radiation pattern
ax0 = fig.add_subplot(2, 2, 1, polar=True)
ax0.set_facecolor('#0a0a1a')
ax0.plot(theta, S_dipole, color='#60a5fa', lw=2, label='Dipole')
ax0.plot(theta, np.ones_like(theta)*0.5, color='#94a3b8', lw=1, linestyle='--', label='Isotropic')
ax0.set_title('Radiation Pattern
(polar: $\propto \sin^2\theta$)', color='white', fontsize=11, pad=15)
ax0.tick_params(colors='#94a3b8', labelsize=8)
ax0.set_facecolor('#0a0a1a')
ax0.legend(loc='upper right', facecolor='#1e293b', labelcolor='white', fontsize=8)

# 2D Far-field radiation map
for ax in [axes[0,1], axes[1,0], axes[1,1]]:
    ax.set_facecolor('#0a0a1a')
    for sp in ax.spines.values(): sp.set_edgecolor('#334155')

r_arr = np.linspace(0.5*lam, 8*lam, 200)
theta_arr = np.linspace(0, np.pi, 200)
R, TH = np.meshgrid(r_arr, theta_arr)
X = R * np.sin(TH)
Z = R * np.cos(TH)

# Far-field: E_theta ~ sin(theta)/r * cos(omega*t - kr)
k = omega/c
t0 = 0
p0 = q * d  # dipole moment amplitude
# Far-field amplitude: |E| ~ (p0*omega^2)/(4pi*eps0*c^2) * sin(theta)/r
E_ff = np.sin(TH) / R * np.cos(k*R - omega*t0)
S_ff = E_ff**2  # Poynting ∝ E²

im1 = axes[0,1].pcolormesh(X/lam, Z/lam, S_ff, cmap='hot', shading='auto',
                            vmin=0, vmax=np.percentile(S_ff, 98))
plt.colorbar(im1, ax=axes[0,1], label=r'$S propto E^2$')
axes[0,1].set_xlabel(r'$x/lambda$', color='#94a3b8'); axes[0,1].set_ylabel(r'$z/lambda$', color='#94a3b8')
axes[0,1].set_title('Far-field Intensity (dipole)', color='white', fontsize=11)
axes[0,1].tick_params(colors='#94a3b8')
axes[0,1].plot(0, 0, 'w*', ms=12, label='Source')
axes[0,1].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)

# Retardation: signal arrives at r with delay r/c
r_obs = 3*lam   # observer at 3 wavelengths
t_obs = np.linspace(0, 5/f, 1000)
# Source position at retarded time t_ret = t - r/c
z_source = d * np.cos(omega * (t_obs - r_obs/c))
E_obs = np.cos(omega*(t_obs - r_obs/c))
axes[1,0].plot(t_obs*f, z_source*100, color='#60a5fa', lw=2, label='Source z(t_ret)')
ax_twin = axes[1,0].twinx()
ax_twin.plot(t_obs*f, E_obs, color='#f59e0b', lw=2, alpha=0.8, label='E_obs(t)')
ax_twin.tick_params(colors='#94a3b8')
ax_twin.set_ylabel('E [arb]', color='#f59e0b')
axes[1,0].set_xlabel('t × f (periods)', color='#94a3b8')
axes[1,0].set_ylabel('z_source [cm]', color='#60a5fa')
axes[1,0].set_title(f'Retarded fields: delay = r/c = {r_obs/c*f:.1f} periods', color='white', fontsize=11)
axes[1,0].tick_params(colors='#94a3b8')
axes[1,0].grid(alpha=0.2, color='#334155')

# Power radiated vs frequency (Larmor)
f_arr = np.linspace(1e6, 1e10, 500)
omega_arr = 2*np.pi*f_arr
eps0 = 8.854e-12
# Larmor: P = q^2*a^2/(6pi*eps0*c^3), a = d*omega^2
P_arr = q**2 * (d*omega_arr**2)**2 / (6*np.pi*eps0*c**3)
axes[1,1].loglog(f_arr/1e6, P_arr*1e12, color='#34d399', lw=2)
axes[1,1].set_xlabel('f [MHz]', color='#94a3b8'); axes[1,1].set_ylabel('P [pW]', color='#94a3b8')
axes[1,1].set_title(r'Larmor Power $P propto omega^4 propto f^4$', color='white', fontsize=11)
axes[1,1].tick_params(colors='#94a3b8')
axes[1,1].grid(alpha=0.2, color='#334155')
axes[1,1].axvline(f/1e6, color='#f59e0b', lw=1.5, linestyle='--', label=f'{f/1e6:.0f} MHz')
axes[1,1].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)

plt.tight_layout()
plt.savefig('lienard_wiechert.png', dpi=130, bbox_inches='tight', facecolor=fig.get_facecolor())
eps0 = 8.854e-12
P = q**2*(d*omega**2)**2/(6*np.pi*eps0*c**3)
print(f"Radiated power at {f/1e6:.0f} MHz: P = {P*1e15:.3f} fW")

Click Run to execute the Python code

First run will download Python environment (~15MB)

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