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A Derivation of the Proposition that an Accelerated Charge Radiates Electromagnetic Waves |
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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
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
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
The vector potential A is such that B is equal to its curl; i.e.,
Conservation of charge requires that
The Biot-Savart Law for the magnetic induction B due to a current of I in a long straight wire is given by
where R is the distance from the point of observation to the wire.
Ampere's Law can be expressed at a point as
or in integral form as
where I stands for current and C denotes a closed curve.
The scalar and vector potentials for a point charge of magnitude e are
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
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'
The potential function given above are known as the Liénard-Wiechert potentials. From them can be derived
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
where E_{a} is the component of the electric field in the direction of the acceleration of the electrical charge. That component is given by
Note that n×(n×(dβ/dt) is in the direction of (dβ/dt), the acceleration.
(To be continued.)
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