Radiation from Accelerating Charges

A non-relativistic charge $q$ undergoing acceleration $\mathbf{a}$ radiates power:

$P = \frac{q^2 a^2}{6\pi\varepsilon_0 c^3}$

For an oscillating dipole $\mathbf{p} = q\mathbf{d}\cos\omega t$ with acceleration $a = \omega^2 d$:

$P = \frac{q^2 \omega^4 d^2}{12\pi\varepsilon_0 c^3} = \frac{\omega^4 p_0^2}{12\pi\varepsilon_0 c^3}$

Where $p_0 = qd$ is the dipole moment amplitude.

Radiation resistance: Equating $P = \frac{1}{2}I_0^2 R_{\text{rad}}$ for an antenna of length $\ell$ carrying current $I_0$ at frequency $\omega$:

$R_{\text{rad} = \frac{\mu_0 c}{6\pi}\left(\frac{\omega \ell}{c}\right)^2 = \frac{\pi}{6}Z_0\left(\frac{\ell}{\lambda}\right)^2 \approx 197\left(\frac{\ell}{\lambda}\right)^2\ \Omega}$

The radiation fields from an oscillating electric dipole at distance $r \gg \lambda$:

$\mathbf{E} = -\frac{\mu_0 \omega^2 p_0}{4\pi r}\sin\theta\, e^{i(kr - \omega t)}\,\hat{\boldsymbol{\theta}}$

$\mathbf{B} = -\frac{\mu_0 \omega^2 p_0}{4\pi c\, r}\sin\theta\, e^{i(kr - \omega t)}\,\hat{\boldsymbol{\phi}}$

The angular distribution of radiated power:

$\frac{dP}{d\Omega} = \frac{\mu_0 p_0^2 \omega^4}{32\pi^2 c}\sin^2\theta$

The total power (integrating over solid angle):

$P = \frac{\mu_0 p_0^2 \omega^4}{12\pi c}$

The radiation pattern is toroidal (doughnut-shaped), with zero radiation along the dipole axis ($\theta = 0, \pi$) and maximum in the equatorial plane ($\theta = \pi/2$).

For a relativistic charge with velocity $\boldsymbol{\beta} = \mathbf{v}/c$ and acceleration $\dot{\boldsymbol{\beta}}$:

$P = \frac{q^2}{6\pi\varepsilon_0 c}\gamma^6\left[(\dot{\boldsymbol{\beta}})^2 - (\boldsymbol{\beta} \times \dot{\boldsymbol{\beta}})^2\right]$

For linear acceleration ($\boldsymbol{\beta} \parallel \dot{\boldsymbol{\beta}}$):

$P = \frac{q^2}{6\pi\varepsilon_0 c}\gamma^6\dot{\beta}^2$

For circular acceleration ($\boldsymbol{\beta} \perp \dot{\boldsymbol{\beta}}$E.g., synchrotron):

$P = \frac{q^2}{6\pi\varepsilon_0 c}\gamma^4\dot{\beta}^2 = \frac{q^2 c}{6\pi\varepsilon_0}\frac{\gamma^4}{R^2}$

Where $R$ is the radius of curvature. The $\gamma^4$ factor (vs. $\gamma^6$ for linear) explains why synchrotron radiation is significant for relativistic electrons but negligible for protons at the same energy ($\gamma$ is $m_p/m_e \approx 1836$ times smaller).

Synchrotron radiation spectrum: The critical frequency is $\omega_c = \frac{3}{2}\gamma^3\frac{c}{R}$. The spectrum peaks near $\omega_c$ and extends to high harmonics, making synchrotron radiation a powerful broadband source from infrared to X-rays.

• The Larmor formula is the non-relativistic limit of the full relativistic expression.
• The angular distribution $\sin^2\theta$ implies zero radiation along the acceleration axis.
• Synchrotron radiation power scales as $\gamma^4$ for circular motion but $\gamma^6$ for linear motion.
• The critical frequency $\omega_c \propto \gamma^3$ means higher energy electrons produce higher frequency radiation.

• Confusing the radiated power with the energy density of the field. The power is the flux of the Poynting vector over a sphere, not the field energy.
• Forgetting the $\gamma^4$ vs $\gamma^6$ distinction when comparing synchrotron and linear acceleration radiation.
• Using the non-relativistic Larmor formula at relativistic speeds without applying the appropriate Lorentz transformation.
• Neglecting that the radiation reaction force is small compared to the Lorentz force for most practical accelerator configurations.

• Synchrotron light sources: Produce intense broadband radiation from infrared to X-rays for materials science, biology, and chemistry experiments.
• Astrophysics: Explains radiation from pulsars, active galactic nuclei, and cosmic microwave background fluctuations.
• Bremsstrahlung: X-ray production in medical imaging and industrial inspection uses radiation from decelerating electrons.
• Antenna theory: The radiation resistance formula guides the design of dipole and monopole antennas for communication systems.

Problem. An electron decelerates from $v = 0.1c$ to rest in a distance $d = 1$ mm inside a metal target. Estimate the fraction of kinetic energy radiated as bremsstrahlung.