Solution

The consistent Liénard–Wiechert potential obeys

A = u φ / c^{2}

. To first order in

β = u / c

, its large-distance form is

φ ≃ \frac{q}{4 π ϵ _{0} r} [1 + n \cdot β (t_{r})], A ≃ \frac{μ _{0} q}{4 π r} u (t_{r}), t_{r} ≃ t - \frac{r}{c} .

Terms varying as

1/ r^{2}

give near fields and are negligible for the radiated power at large

r

. Keep the first-order velocity term in

φ

: although small, its retarded-time derivative contributes a leading acceleration field. To the required order,

\partial t_{r} / \partial t ≃ 1

and

\nabla t_{r} ≃ - n / c

. Holding the observation direction fixed at leading order gives

- \nabla φ_{1/ r} = \frac{q}{4 π ϵ _{0} cr} n (n \cdot \dot{β}), - \frac{\partial A}{\partial t}_{1/ r} = - \frac{q}{4 π ϵ _{0} cr} \dot{β} .

Thus the outgoing electric field is transverse:

E_{rad} (r, t) = \frac{q}{4 π ϵ _{0} cr} [n \times (n \times \dot{β})]_{t_{r}} = \frac{q}{4 π ϵ _{0} c ^{2} r} [n \times (n \times a)]_{t_{r}} .

(3)

Here

a = d u / d t_{r}

. The magnetic field follows from

\nabla \times A

, or from the outgoing plane electromagnetic wave relation:

B_{rad} = - \frac{μ _{0} q}{4 π r} [n \times \dot{β}]_{t_{r}} = \frac{1}{c} n \times E_{rad} .

(4)

The minus sign is essential. Both fields fall as

1/ r

, are mutually orthogonal and obey

∣ E ∣ = c ∣ B ∣

. The corresponding Liénard-Wiechert field also contains nonradiative terms, omitted in this leading far-field calculation.

Mean angular power

Since

E \cdot n = 0

, the Poynting vector is

S = \frac{1}{μ _{0}} E \times B = ϵ_{0} c ∣ E ∣^{2} n .

The area on a distant sphere corresponding to solid angle

d Ω

r^{2} d Ω

. Consequently

\frac{d ⟨ P ⟩}{d Ω} = \frac{q ^{2}}{16 π ^{2} ϵ _{0} c ^{3}} ⟨ ∣ a ∣^{2} - (n \cdot a)^{2} ⟩ = \frac{q ^{2}}{16 π ^{2} ϵ _{0} c} ⟨ ∣ n \times \dot{β} ∣^{2} ⟩ .

The average is over the source motion at retarded time. At leading nonrelativistic order, differences between source-time and reception-time averaging are beyond the approximation. The units are power per solid angle,

W s r^{- 1}

The acceleration covariance in nonrelativistic radiation is another way to organize this result: with

C_{ij} = ⟨ a_{i} a_{j} ⟩

, the angular factor is

tr C - n^{T} C n

. Integrating over all observation directions uses

\int [∣ a ∣^{2} - (n \cdot a)^{2}] d Ω = (8 π /3) ∣ a ∣^{2}

and gives the averaged Larmor formula,

⟨ P ⟩ = q^{2} ⟨ a^{2} ⟩ / (6 π ϵ_{0} c^{3})

Linear oscillation

For the prescribed displacement along

z

, the source acceleration is

a (t_{r}) = - a ω^{2} cos (ω t_{r}) z

. If

θ

is the angle from the

z

axis,

∣ a ∣^{2} - (n \cdot a)^{2} = a^{2} ω^{4} cos^{2} (ω t_{r}) sin^{2} θ

. Averaging

cos^{2}

over one wave period gives

\frac{d ⟨ P ⟩ _{linear}}{d Ω} = \frac{q ^{2} a ^{2} ω ^{4}}{32 π ^{2} ϵ _{0} c ^{3}} sin^{2} θ .

The angular distribution of electric dipole radiation has zero intensity on the oscillation axis and maximum intensity in the equatorial plane, with no azimuthal dependence. Its total power is

⟨ P ⟩_{linear} = \frac{q ^{2} a ^{2} ω ^{4}}{12 π ϵ _{0} c ^{3}} .

The condition

aω ≪ c

is the required nonrelativistic condition. The plotted radial coordinate below is angular power, rather than field wave amplitude.

Circular orbit

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 .