= Radiation pattern of a circularly moving nonrelativistic charge
{title2=$d\langle P\rangle/d\Omega=q^2b^2\omega^4(1+\cos^2\theta)/(32\pi^2\epsilon_0c^3)$}
For an orbit of <radius> $b$ in the $xy$ plane, $\mathbf a=-b\omega^2(\cos\omega t,\sin\omega t,0)$. In <spherical coordinates>, at <angle> $\theta$ from the orbit normal, $\langle(\mathbf n\cdot\mathbf a)^2\rangle=b^2\omega^4\sin^2\theta/2$. The <acceleration covariance in nonrelativistic radiation> therefore gives the displayed pattern. Its axial <power> per <solid angle> is twice its equatorial value; no observation direction has zero time-averaged <power>. Integrating gives $\langle P\rangle=q^2b^2\omega^4/(6\pi\epsilon_0c^3)$. The condition $b\omega\ll c$ is essential: relativistic <synchrotron radiation> has a different beamed pattern.
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