When is Gauss' law not applicable?

Question

Does Gauss law hold in all cases, including the ones where charges are moving at speeds comparable to light?

What I think: Gauss law does not hold true when charges are moving at very high speeds. For example, assume a sphere and a charge kept midway between the center and a point on the sphere. If the charge moves towards A, then, the increase in electric field is communicated to the region near A quicker than the decrease in the field near the diametrically opposite end of A is communicated, so the flux should be more than expected at that time. Obviously, this is just a thought and may be wrong.

Does this answer your question? Gauss's Law with Moving Charges — fqq, Dec 14 '20 at 18:49
The post marked as a duplicate addresses the same question but the answer there is incomplete as it does not account for time delay, as explicitly requested by the OP. — my2cts, Dec 14 '20 at 21:12

score 7 · Answer 1 · answered Dec 14 '20 at 18:49

7

Gauss’ Law is the integral form of one of Maxwell’s equations,

$$\nabla\cdot\mathbf E=\frac{\rho}{\epsilon_0}.$$

All of Maxwell’s equations are consistent with Special Relativity; they don’t change their form under Lorentz transformations.

Therefore Gauss’ Law applies in all inertial reference frames, including ones where charged particles are moving at speeds near $c$.

answered Dec 14 '20 at 18:49

G. Smith

51,534

The question is if the integral form of Gauss's law is covariant. – my2cts Dec 14 '20 at 19:26
1

The integral forms of Maxwell’s equations are mathematically equivalent to the differential forms, so it is. – G. Smith Dec 14 '20 at 20:04

When is Gauss' law not applicable?

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