Classical Electromagnetism - General Field Equations

General Field Equations

As simple and satisfying as Coulomb's equation may be, it is not entirely correct in the context of classical electromagnetism. Problems arise because changes in charge distributions require a non-zero amount of time to be "felt" elsewhere (required by special relativity).

For the fields of general charge distributions, the retarded potentials can be computed and differentiated accordingly to yield Jefimenko's Equations.

Retarded potentials can also be derived for point charges, and the equations are known as the Liénard-Wiechert potentials. The scalar potential is:

\varphi = \frac{1}{4 \pi \varepsilon_0} \frac{q}{\left| \mathbf{r} - \mathbf{r}_q(t_{ret}) \right|-\frac{\mathbf{v}_q(t_{ret})}{c} \cdot (\mathbf{r} - \mathbf{r}_q(t_{ret}))}

where q is the point charge's charge and r is the position. rq and vq are the position and velocity of the charge, respectively, as a function of retarded time. The vector potential is similar:

\mathbf{A} = \frac{\mu_0}{4 \pi} \frac{q\mathbf{v}_q(t_{ret})}{\left| \mathbf{r} - \mathbf{r}_q(t_{ret}) \right|-\frac{\mathbf{v}_q(t_{ret})}{c} \cdot (\mathbf{r} - \mathbf{r}_q(t_{ret}))}.

These can then be differentiated accordingly to obtain the complete field equations for a moving point particle.

Read more about this topic:  Classical Electromagnetism

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