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

Famous quotes containing the words general and/or field:

    “Surely one of the peculiar habits of circumstances is the way they follow, in their eternal recurrence, a single course. If an event happens once in a life, it may be depended upon to repeat later its general design.”
    —Ellen Glasgow (1873–1945)

    “... many American Jews have a morbid tendency to exaggerate their handicaps and difficulties. ... There is no doubt that the Jew ... has to be twice as good as the average non- Jew to succeed in many a field of endeavor. But to dwell upon these injustices to the point of self-pity is to weaken the personality unnecessarily. Every human being has handicaps of one sort or another. The brave individual accepts them and by accepting conquers them.”
    —Agnes E. Meyer (1887–1970)