﻿ A Derivation of the Proposition that an Accelerated Charge Radiates Electromagnetic Waves
San José State University

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Thayer Watkins
Silicon Valley
U.S.A.

A Derivation of the Proposition that
Electromagnetic Waves

## Electrical and Magnetic Fields

Electrical and magnetic fields are usually represesnted by the electrical field and magnetic induction field intensity vectors, E and B respectively. There are other equivalent representations such as the displacement field and magnetic field, D and H respectively, where

#### D = εE and H = (1/μ)B

where ε and μ are scalar quantities characteristic of the material called, respectively, the dialectic and the permability.

Some advantage is gained in terms of the simplicity of the equations by using D and H as well as E and B. The field equations are

#### ∇·D = 4πρ ∇×E = 0 ∇·B = 0 ∇×H = 4πJ/c

where ρ is the charge density, J is the current flow intensity and c is the speed of light in in a vacuum. ∇·B must be zero because there are no such things as magnetic charges. ∇×E must be zero because there cannot be a flow of magnetic charges since magnetic charges do not exist.

An alternate representation of the electric and magnetic fields is in terms of potential functions. Let Φ be a scalar field such that E is the negative of its gradient; i.e.,

Then

#### ∇²Φ = −4πρ

The vector potential A is such that B is equal to its curl; i.e.,

## The Laws of Electrodynamics

Conservation of charge requires that

#### (∂ρ/∂t) + ∇·J = 0

The Biot-Savart Law for the magnetic induction B due to a current of I in a long straight wire is given by

#### B = 2I/(cR)

where R is the distance from the point of observation to the wire.

Ampere's Law can be expressed at a point as

#### ∇×H = 4πJ/c

or in integral form as

#### ∫CB·ds = 4πI/c

where I stands for current and C denotes a closed curve.

## The Fields of a Point Charge

The scalar and vector potentials for a point charge of magnitude e are

#### Φ(x, t) = e[1/(κR)]ret A(x, t) = e[β/(κR)]ret

where β is the velocity vector of the particle relataive to the speed of light; i.e., β=v/c. The constant κ depends upon the system of units used. The variable R is the distance from the point of observation x to the location of the point charge x'; i.e,, R = |(x−x'|. R at time t is based upon where the point charge was at time t', where

#### t' = t − R(t')/c

This t' is known as retarded time. It is not always a trivial matter to determine t' from t. For a linear coordinate system with x>x' and x'=vt'

#### t' = t − (x−x')/c and hence t' − (v/c)t' = t − x/c and therefore t' = (t − x/c)/(1 − v/c)

The potential function given above are known as the Liénard-Wiechert potentials. From them can be derived

#### E(x, t) = e[(n-β)(1-β²)/(κ³R²)]ret + (e/c)[(n/(κ³R)×(n-β)×(dβ/dt)]retand B = n×E

where β is the vector of velocity relative to the speed of light; i.e., β=v/c. The vector n is a unit vector in the direction from the position of the particle to the point of observation.

The energy flux is given by the Poynting vector

#### S = (c/4π)(E×H) which reduces to S = (c/4π)|Ea|²n

where Ea is the component of the electric field in the direction of the acceleration of the electrical charge. That component is given by

#### Ea = (e/c)[(n×(n×(dβ/dt)/R]ret

Note that n×(n×(dβ/dt) is in the direction of (dβ/dt), the acceleration.

(To be continued.)