3Magnetostatics

IB Electromagnetism

3 Magnetostatics

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

E

= 0. The remaining Maxwell’s equations

are

∇ ×B = µJ

∇ · B = 0

The objective is, given a J, find the resultant B.

Before we start, what does the condition

ρ

= 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-

tion:

∂ρ

∂t

+ ∇ · 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.