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Liénard-Wiechert field 2026-10-18

The Liénard-Wiechert field is the electric field and magnetic field obtained from the retarded potentials of a moving point charge. In the nonrelativistic radiation zone, the electric radiation field is

E_{rad} = q n \times (n \times a) / (4 π ϵ_{0} c^{2} R)

at retarded time. Its outgoing Poynting vector integrates to the Larmor formula. Coulomb and induction fields must also be retained when a near-field calculation is required.

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

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}} .

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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Past exam of the optics and electrodynamics course of the University of Cambridge / 2021 / Section A / A3 / Solution 2026-10-18

After field electron emission, let

h

denote the plate separation of the parallel-plate capacitor, which the question does not specify. The uniform capacitor electric field has magnitude

E_{c} = ∣ V ∣/ h

, and a nonrelativistic Electron of mass

m_{e}

has constant acceleration magnitude

a = e E_{c} / m_{e}

. The sign of its charge affects the direction of the radiation field but not its radiated power.

In the supplied nonrelativistic Liénard-Wiechert field,

\dot{β} = a / c

. At polar angle

θ

to the acceleration,

∣ n \times (n \times a) ∣ = a sin θ

, hence

E_{rad} = \frac{e a s i n θ}{4 π ϵ _{0} c ^{2} R} .

The outgoing Poynting vector has magnitude

S = ϵ_{0} c E_{rad}^{2}

. Its radiated power per solid angle is therefore

\frac{d P}{d Ω} = R^{2} S = \frac{e ^{2} a ^{2}}{16 π ^{2} ϵ _{0} c ^{3}} sin^{2} θ .

Using

\int sin^{2} θ d Ω = 8 π /3

gives the Larmor formula and the requested constant power during the uniform-field part of the flight:

P = \frac{e ^{2} a ^{2}}{6 π ϵ _{0} c ^{3}} = \frac{e ^{4} E _{c}^{2}}{6 π ϵ _{0} m _{e}^{2} c ^{3}} = \frac{e ^{4} V ^{2}}{6 π ϵ _{0} m _{e}^{2} c ^{3} h ^{2}} .

The brackets in the given field require the acceleration at retarded time; the radiation reaches the far observer after the propagation delay. This derivation assumes that radiation reaction is a small correction to the acceleration and that the flight is nonrelativistic, for example

e ∣ V ∣ ≪ m_{e} c^{2}

for release from rest. Entry, emission and impact transients are not described by the uniform-field model. A result solely in terms of

V

cannot be obtained without a plate spacing or a specified internal field.

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