Maxwell raised the question: what are the magnetic fields produced by non-steady currents? He noticed that Ampere's Law contradicts the conservation of charge and required modification when currents are not steady. These two laws, as we have noted, correspond to the equations
cB = j
Taking the divergence of both sides of the latter and using the former gives:
0 =B =j = -/ t
Thus, Ampere's Law implies that the current in it is steady.
Maxwell noted that the electrostatic equation,E =, provides a way to modify Ampere's Law so that it is consistent with non-steady currents. If we add a term E / t to its left hand side, we can eliminate the condition that be constant, and obtain a consistent set of equations: These equations are called Maxwell's Equations.
B +E / ct = j / c (Maxwell's modified Ampere's Law)
B = 0 (No magnetic sources or sinks)
E -B / ct = 0 (Faraday's Law)
E = (Gauss's Theorem)