11.4 Magnetostatics: the field of steady currents

Steady currents produce magnetic fields, and magnetic fields exert forces on moving charges. Magnetostatics is the magnetic counterpart of electrostatics, with two structural differences: the field circulates around its sources rather than diverging from them, and there are no magnetic charges for it to diverge from at all.

The Lorentz force

The magnetic field $\mathbf{B}$ is defined by the force it exerts on a moving charge. Together with the electric force, the total force on a charge $q$ moving at velocity $\mathbf{v}$ is the Lorentz force:

\mathbf{F} \;=\; q\,(\mathbf{E} + \mathbf{v}\times\mathbf{B}).

where

$\mathbf{F}$: force on the charge N
$q$: charge C
$\mathbf{v}$: charge velocity m/s
$\mathbf{B}$: magnetic field T

The magnetic part acts perpendicular to the velocity, so it does no work — it bends trajectories into circles and helices but never changes a charge’s speed. This is why magnetic forces steer particles in accelerators and confine plasmas without heating them.

No magnetic charge

No isolated magnetic pole has ever been found: field lines of $\mathbf{B}$ form closed loops, never beginning or ending. The differential statement is the second Maxwell equation,

\nabla\cdot\mathbf{B} \;=\; 0,

the exact magnetic analogue of $\nabla\cdot\mathbf{E} = \rho/\varepsilon_0$ but with the source set permanently to zero. Where electric field lines spring from charges, magnetic field lines only circulate.

Ampère’s law and Biot–Savart

What magnetic fields circulate around is current. Ampère’s law states that the circulation of $\mathbf{B}$ around any closed loop equals the enclosed current,

\oint \mathbf{B}\cdot d\boldsymbol{\ell} \;=\; \mu_0 I_\text{enc}, \qquad \nabla\times\mathbf{B} \;=\; \mu_0\mathbf{J},

where

$\mu_0$: permeability of free space H/m
$I_\text{enc}$: current threading the loop A
$\mathbf{J}$: current density A/m²

the magnetic mirror of Gauss’s law: where Gauss relates the flux of $\mathbf{E}$ to enclosed charge, Ampère relates the circulation of $\mathbf{B}$ to enclosed current. As with Gauss’s law, symmetry turns it into a direct computation.

▶ The field of a long straight wire Derivation

For an infinite straight wire carrying current $I$ , symmetry makes $\mathbf{B}$ circle the wire with constant magnitude at radius $r$ . Ampère’s law on a circle of radius $r$ gives $B\,(2\pi r) = \mu_0 I$ , so

B = \frac{\mu_0 I}{2\pi r}.

The field circles the wire and falls as $1/r$ — the same radial dependence Gauss’s law gives for a line of charge, now wrapped around the current instead of pointing away from it.

When symmetry is absent, the field is built up from current elements by the Biot–Savart law, $d\mathbf{B} = (\mu_0/4\pi)\,I\,d\boldsymbol{\ell}\times\hat{\mathbf{r}}/r^2$ — the magnetic counterpart of the Coulomb superposition integral. With the three static equations now in hand — $\nabla\cdot\mathbf{E} = \rho/\varepsilon_0$ , $\nabla\cdot\mathbf{B} = 0$ , $\nabla\times\mathbf{B} = \mu_0\mathbf{J}$ — only the coupling between the fields remains, and that coupling appears when the fields change in time.