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 Incoming links: Radiation pattern of a circularly moving nonrelativistic charge

Past exam of the optics and electrodynamics course of the University of Cambridge / 2019 / Section B / B4 / Circular orbit 2026-10-18

Denote the orbit radius by

b

here to distinguish it from observation distance; it is the quantity called

R

in this part of the paper. Write

x (t_{r}) = b (cos ω t_{r}, sin ω t_{r}, 0), a (t_{r}) = - b ω^{2} (cos ω t_{r}, sin ω t_{r}, 0) .

For observation at spherical coordinates

(θ, ϕ)

, measured from the orbit normal,

n \cdot a = - b ω^{2} sin θ cos (ω t_{r} - ϕ), ⟨(n \cdot a)^{2} ⟩ = \frac{1}{2} b^{2} ω^{4} sin^{2} θ .

The acceleration magnitude is constant,

∣ a ∣ = b ω^{2}

. Therefore the radiation pattern of a circularly moving nonrelativistic charge is

\frac{d ⟨ P ⟩ _{circular}}{d Ω} = \frac{q ^{2} b ^{2} ω ^{4}}{32 π ^{2} ϵ _{0} c ^{3}} (1 + cos^{2} θ), ⟨ P ⟩_{circular} = \frac{q ^{2} b ^{2} ω ^{4}}{6 π ϵ _{0} c ^{3}} .

Time averaging removes the azimuthal dependence. The axial intensity is twice the equatorial intensity, and no direction has zero mean intensity: the instantaneous dipole axis continually rotates. This is also the sum of the two orthogonal linear-oscillation patterns with their relative phase quadrature. At equal radius and linear-oscillation amplitude, the total mean power is twice the linear result. The calculation requires

bω ≪ c

; relativistic synchrotron radiation is outside this approximation.

Figure 1.
Normalized angular power for linear oscillation and nonrelativistic circular motion, with angle measured from the same z axis
. Details.

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