Divergence of Electric Field of Continuous and of a Point Charge for Relativistic and non-Relativistic Motion

TL;DR

This paper investigates how the divergence of electric and magnetic fields behaves for different charge distributions and motions, revealing potential modifications to Gauss's law for time-varying distributed charges.

Contribution

It demonstrates that divergence of electric field can be non-zero in regions without charges for time-varying continuous distributions, suggesting possible modifications to Gauss's law.

Findings

Divergence of electric field is non-zero in charge-free regions for time-varying distributions.

Divergence of magnetic field remains zero across all cases.

Time variation of charges must be very rapid for effects to be experimentally observable.

Abstract

In this paper we considered divergence of electric and of magnetic fields for four cases: classical point charge, classical continuous charge, relativistic point and relativistic continuous charges. Results for classical and relativistic point charges are the same as in literature, i.e. Gauss's law is valid. However results for time-varying classical and relativistic distributed charges indicate that divergence of electric field is not zero even for volumes of space where no charges are present. For these cases original Gauss's law might require modification. Divergence of electric field seems to be far-field type scalar anisotropic field, which is generated by time-varying electric charges or currents. Results indicate that for these effects to be sufficiently large to be experimentally observable the time variation of electric charges and/or of currents should be very fast. Divergence of magnetic field is zero for all cases.