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A Derivation of the Proposition that
an Accelerated Charge Radiates
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
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.,

E = −∇Φ


∇²Φ = −4πρ

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

B = ∇×A

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)]ret
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.)

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