Chapter 16

Larmor Formula & Radiation Reaction

16.1 Larmor Radiation Formula

The total power radiated by a non-relativistic accelerating charge is given by the Larmor formula, derived from the Liénard–Wiechert radiation fields:

$$\boxed{P = \frac{\mu_0 q^2 a^2}{6\pi c} = \frac{q^2 a^2}{6\pi\epsilon_0 c^3}} \quad \text{(Larmor, non-relativistic)}$$

where $a = |\dot{\mathbf{v}}|$ is the magnitude of the acceleration. The angular distribution is:

$$\frac{dP}{d\Omega} = \frac{\mu_0 q^2 a^2}{16\pi^2 c}\sin^2\theta$$

Maximum radiation is emitted perpendicular to the acceleration direction; zero along $\hat{a}$.

16.1.1 Relativistic Generalization (Liénard)

For a relativistic particle (speed $v$, Lorentz factor $\gamma$), the Liénard formula gives:

$$P = \frac{\mu_0 q^2 c}{6\pi} \gamma^6 \left[\left(\frac{\dot{v}}{c}\right)^2 - \left(\frac{\mathbf{v}\times\dot{\mathbf{v}}}{c^2}\right)^2\right]$$

For circular motion (synchrotron radiation), $\mathbf{v} \perp \dot{\mathbf{v}}$:

$$P_{\rm sync} = \frac{\mu_0 q^2 c}{6\pi} \gamma^4 \left(\frac{v}{R}\right)^2 \propto \gamma^4$$

The $\gamma^4$ enhancement makes synchrotron radiation dominant for ultra-relativistic particles. Synchrotron light sources use this to generate intense X-rays.

16.2 Radiation Reaction

A radiating charge loses energy, so there must be a back-reaction force on the charge itself. The Abraham–Lorentz force (radiation reaction) is:

$$\mathbf{F}_{\rm rad} = \frac{\mu_0 q^2}{6\pi c}\dot{\mathbf{a}} = m\tau_0\dot{\mathbf{a}}$$

where the characteristic time is:

$$\tau_0 = \frac{\mu_0 q^2}{6\pi mc} = \frac{q^2}{6\pi\epsilon_0 mc^3} \approx 6.3 \times 10^{-24}\,\text{s (electron)}$$

This is tiny, justifying the non-relativistic approximation for most laboratory scenarios. However, the Abraham–Lorentz equation has pathological solutions (runaway acceleration), a fundamental problem in classical electrodynamics resolved only by quantum electrodynamics.

Classical Hydrogen Atom Problem

By the Larmor formula, an electron orbiting a proton radiates energy continuously. Calculating the spiral collapse time gives $\sim 10^{-11}$ s — the hydrogen atom should collapse almost instantly! This failure of classical electrodynamics to explain atomic stability was one of the key motivations for quantum mechanics.

Simulation: Larmor Formula & Synchrotron Power

Larmor & Synchrotron Radiation

Computes cyclotron radiation power, relativistic γ⁴ enhancement, angular distribution, and classical H-atom collapse time.

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

# ──────────────────────────────────────────────
# Larmor formula: synchrotron-like radiation
# Electron in a circular orbit (cyclotron motion)
# Power radiated vs. radius, compare classical/relativistic
# ──────────────────────────────────────────────
c = 3e8
me = 9.109e-31   # electron mass [kg]
qe = 1.602e-19   # electron charge [C]
eps0 = 8.854e-12
mu0 = 4*np.pi*1e-7

# Larmor formula (non-relativistic): P = q^2 a^2 / (6 pi eps0 c^3)
def larmor_power(q, a):
    return q**2 * a**2 / (6 * np.pi * eps0 * c**3)

# Cyclotron: r = mv/(qB), a = v^2/r = qvB/m
B = 1.0   # 1 Tesla
v_arr = np.linspace(0.01*c, 0.99*c, 400)
gamma_arr = 1/np.sqrt(1 - (v_arr/c)**2)

# Classical (non-relativistic Larmor)
r_class = me * v_arr / (qe * B)
a_class = v_arr**2 / r_class
P_class = larmor_power(qe, a_class)

# Relativistic Larmor (Liénard): P = q²a²γ⁴ / (6πε₀c³) for circular motion
P_rel = qe**2 * a_class**2 * gamma_arr**4 / (6*np.pi*eps0*c**3)

print("Cyclotron radiation in B = 1 T:")
print(f"  v=0.1c: P_class={larmor_power(qe,(0.1*c)**2/(me*0.1*c/(qe*B)))*1e15:.3f} fW")
print(f"  v=0.9c: P_rel (synchrotron) ~ {qe**2*(0.9*c)**2/((me*0.9*c/(qe*B)))**2*((1/np.sqrt(1-0.81))**4)/(6*np.pi*eps0*c**3)*1e12:.3f} pW")

# Radiation from H atom (classical — Bohr orbit problem)
a0 = 5.292e-11   # Bohr radius
v_bohr = qe**2/(4*np.pi*eps0*me*a0*c) * c  # Bohr velocity
a_bohr = v_bohr**2/a0
P_bohr = larmor_power(qe, a_bohr)
t_collapse = me*v_bohr**2/(2*P_bohr)  # classical collapse time ~ 10⁻¹¹ s
print(f"Classical H atom: P = {P_bohr:.3e} W, collapse time = {t_collapse:.3e} s")

# ── Angular distribution ── dP/dΩ = (q²a²)/(16π²ε₀c³) sin²θ
theta_ang = np.linspace(0, 2*np.pi, 500)
dPdOmega = np.sin(theta_ang)**2

# ── Abraham-Lorentz force (radiation reaction) ──
# F_rad = (mu0 q^2)/(6 pi c) * da/dt   (self force)
tau = mu0 * qe**2 / (6*np.pi*me*c)   # characteristic time
print(f"Abraham-Lorentz timescale τ = {tau:.3e} s (≈ 6.3×10⁻²⁴ s)")

fig, axes = plt.subplots(2, 2, figsize=(13, 9))
fig.patch.set_facecolor('#0a0a1a')
ax_polar = fig.add_subplot(2, 2, 1, polar=True)
ax_polar.set_facecolor('#0a0a1a')
ax_polar.plot(theta_ang, dPdOmega, color='#60a5fa', lw=2)
ax_polar.set_title('Angular Distribution
$dP/d\Omega \propto \sin^2\theta$', color='white', fontsize=10, pad=15)
ax_polar.tick_params(colors='#94a3b8', labelsize=7)

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')

beta = v_arr/c
axes[0,1].semilogy(beta, P_class*1e15, color='#60a5fa', lw=2, label='Larmor (non-rel.)')
axes[0,1].semilogy(beta, P_rel*1e15, color='#f59e0b', lw=2, label='Liénard (relativistic)')
axes[0,1].set_xlabel('β = v/c', color='#94a3b8'); axes[0,1].set_ylabel('P [fW]', color='#94a3b8')
axes[0,1].set_title('Radiated Power in B=1T', color='white', fontsize=11)
axes[0,1].tick_params(colors='#94a3b8')
axes[0,1].legend(facecolor='#1e293b', labelcolor='white', fontsize=9)
axes[0,1].grid(alpha=0.2, color='#334155')

# Gamma factor effect
axes[1,0].plot(beta, gamma_arr, color='#34d399', lw=2)
axes[1,0].set_xlabel('β = v/c', color='#94a3b8'); axes[1,0].set_ylabel('γ = Lorentz factor', color='#94a3b8')
axes[1,0].set_title('Lorentz Factor γ(β)', color='white', fontsize=11)
axes[1,0].tick_params(colors='#94a3b8')
axes[1,0].grid(alpha=0.2, color='#334155')
axes[1,0].axhline(1, color='#94a3b8', lw=1, linestyle='--')

# Ratio relativistic/classical
ratio = P_rel/P_class
axes[1,1].semilogy(beta, ratio, color='#a78bfa', lw=2)
axes[1,1].set_xlabel('β = v/c', color='#94a3b8'); axes[1,1].set_ylabel('P_rel / P_class = γ⁴', color='#94a3b8')
axes[1,1].set_title('Relativistic Enhancement γ⁴', color='white', fontsize=11)
axes[1,1].tick_params(colors='#94a3b8')
axes[1,1].grid(alpha=0.2, color='#334155')

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

Click Run to execute the Python code

First run will download Python environment (~15MB)

← Retarded Potentials Course Overview ↑