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:

    “Without metaphor the handling of general concepts such as culture and civilization becomes impossible, and that of disease and disorder is the obvious one for the case in point. Is not crisis itself a concept we owe to Hippocrates? In the social and cultural domain no metaphor is more apt than the pathological one.”
    —Johan Huizinga (1872–1945)

    “When white men were willing to put their own offspring in the kitchen and corn field and allowed them to be sold into bondage as slaves and degraded them as another man’s slave, the retribution of wrath was hanging over this country and the South paid penance in four years of bloody war.”
    —Rebecca Latimer Felton (1835–1930)