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12.4 Ampere's Law

Physicists conclude that electrostatics can be described by the two conditions:E = 0, E =.

No sources or sinks of Magnetic Field lines; and Ampere's Law.

We can immediately translate the statement that magnetic fields have no sources or sinks into the statement:B = 0, since, (again by the divergence theorem,) we know that sources and sinks correspond to non-vanishing divergence.

You are probably familiar with the Biot-Savart Law which describes the magnetic field produced by an electrical  current element.

Ampere's Law states that, for constant current flow, the flux of electrical current through a surface is proportional to the line integral of the magnetic field (counterclockwise) around its boundary:

Stokes' Theorem, applied to B, gives

Upon combining these two we obtain, for any surface S:

from which physicists conclude the differential equations of steady state current flow:

B = j / c, B = 0