 Radiation fields

ID: past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2019/section-b/b4/radiation-fields

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}} .

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} .

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.

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