A Few Tests of Gauss’ Law

G.R.Dixon, 6/12/2005

According to Gauss’ law, the electric field flux through a closed surface is proportional to the enclosed charge. This law presumably applies to point charges as well as to distributions of charge with finite densities. If q is a point charge inside a closed surface, then according to Gauss,

. (1)

In this article Eq. 1 is tested for 3 simple cases: (1) q (1 coulomb in every case) is at rest at points on the x-axis; (2) q moves with a constant velocity along the x-axis; and (3) q oscillates on the x-axis. In every case the spherical surface of integration is centered on the origin and has a radius
R specified by the user. In general F_{E}/(q/e_{o}) is plotted for comparison to unity. In the first two cases F_{E}/(q/e_{o}) is plotted as a function of x (q’s position). In case 3 F_{E}/(q/e_{o}) is plotted as a function of t over the range of an oscillation period.

In cases 2 and 3 the user can specify the constant or maximum speed, and hence both non-relativistic and relativistic cases can be considered. The relativistically rigorous point charge field solutions are used in
computing dF_{E},
which is the flux increment through dA (dA being the area of a circular strip of the surface of integration that is concentric to the x-axis).

In the following 3 plots (Cases 1 and 2) Eq. 1 appears to be essentially satisfied. To the extent Gauss’ law applies, flat plots of unity might have been expected. The slight excursions from unity are probably attributable to numerical errors.

Figure 1

Stationary q, 0__<__x<R/2

Figure 2

Constant Velocity q, v=.01c, 0__<__x<R/2

Figure 3

Constant Velocity q, v=.98c, 0__<__x<R/2

The following three figures plot F_{E}/(q/e_{o}) for an __oscillating__ q, (a) with non-relativistic maximum v = .01c, (b) with mildly relativistic maximum v = .75c, and (c) with highly relativistic maximum v = .98c. The amplitude of oscillation is R/2 The varying shapes are interesting. The flux-computing software indicates the following average values of F_{E}/(q/e_{o}): (a) maxv=.01c: 1.0000195; (b) maxv=.75c: 1.0000195; (c) maxv=.98c: 1.0000202.

**Figure 4**

Oscillating q, maxv=.01c

Figure 5

Oscillating q, maxv=.75c

Figure 6

Oscillating q, maxv=.98c