Advanced Electrodynamics

Manual: General · Subject: Electrodynamics

Radiation, retarded potentials, and multipoles

Radiation from Accelerating Charges

Retarded Potentials and Lienard-Wiechert

In electrodynamics, changes propagate at speed $c$ . The retarded potentials:<br> $\phi(\vec{r},t) = \frac{1}{4\pi\varepsilon_0}\int \frac{\rho(\vec{r}',t_r)}{|\vec{r}-\vec{r}'|}d^3r'$ where $t_r = t - |\vec{r}-\vec{r}'|/c$ is the retarded time.

Liénard-Wiechert potentials give exact fields for a point charge moving arbitrarily.

Radiation Power — Larmor Formula

An accelerating charge $q$ with non-relativistic acceleration $a$ radiates power:<br> $P = \frac{q^2 a^2}{6\pi\varepsilon_0 c^3}$ (Larmor formula). Relativistic generalisation (Liénard): $P = \frac{q^2\gamma^6}{6\pi\varepsilon_0 c}\left(a^2 - |\vec{v}\times\vec{a}|^2/c^2\right)$ .

Multipole radiation: electric dipole $\sim (\ddot{\vec{p}})^2$ , magnetic dipole $\sim (\ddot{\vec{m}})^2$ .

ℹ️

Synchrotron Radiation

Relativistic electrons in circular orbits radiate synchrotron radiation — highly directional, broad-spectrum, and intense. Used in particle accelerators and synchrotron light sources for materials science.

A charge undergoing constant velocity motion (no acceleration) in vacuum:

Does not radiate electromagnetic energy Radiates at all velocities Radiates only above

c/2

Creates only a static electric field

Electric dipole radiation is proportional to:

|\ddot{\vec{p}}|^2

|\vec{p}|^2

|\dot{\vec{p}}|^2

|\vec{p}|

Explain why a circular electron orbit in a classical model of the hydrogen atom is unstable according to classical electrodynamics.

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Advanced Electrodynamics

Radiation from Accelerating Charges

Retarded Potentials and Lienard-Wiechert

Radiation Power — Larmor Formula