Charges give rise to electric fields, currents give rise to magnetic fields. In this
section, we study the magnetic fields induced by steady currents, i.e. in situations
where J 6= 0 and ρ = 0. Again, we look for time-independent solutions.
Since there is no charge, we obtain
= 0. The remaining Maxwell’s equations
∇ ×B = µJ
∇ · B = 0
The objective is, given a J, find the resultant B.
Before we start, what does the condition
= 0 mean? It does not mean that
there are no charges around. We want charges to be moving around to create
current. What it means is that the positive and negative charges balance out
exactly. More importantly, it stays that way. We don’t have a net charge flow
from one place to another. At any specific point in space, the amount of charge
entering is the same as the amount of charge leaving. This is the case in many
applications. For example, in a wire, all electrons move together at the same
rate, and we don’t have charge building up at parts of the circuit.
Mathematically, we can obtain the interpretation from the continuity equa-
+ ∇ · J = 0.
In the case of steady currents, we have ρ = 0. So
∇ · J = 0,
which says that there is no net flow in or out of a point.