Magnetostatic Fields

A stationary charge makes only an electric field. Set it moving — a current — and a new field appears around it: the magnetic field. We describe it with two vectors, just as electrostatics used \(\vec{E}\) and \(\vec{D}\). The magnetic field intensity \(\vec{H}\) (A/m) depends only on the currents; the magnetic flux density \(\vec{B}\) (tesla) includes the medium:

The parallel with electrostatics is exact: \(\vec{H}\) mirrors \(\vec{D}\) (source-only), \(\vec{B}\) mirrors \(\vec{E}\) (medium-dependent), and \(\mu_0\) mirrors \(\varepsilon_0\). Keeping this dictionary in mind makes the whole of Part 3 feel like familiar territory seen in a mirror.

The Biot–Savart law is the magnetic Coulomb's law: it gives the field \(d\vec{H}\) produced by a small current element \(I\,d\vec{l}\). Like Coulomb's law it falls off as \(1/R^2\), but a cross product makes the field circle the current rather than point away from it:

The cross product encodes the right-hand rule: point the thumb along the current and the fingers curl in the direction of \(\vec{H}\). Integrating Biot–Savart along a whole circuit gives the total field — the magnetic counterpart of summing Coulomb contributions.

Just as Gauss's law shortcut the Coulomb integral, Ampère's circuital law shortcuts Biot–Savart when symmetry allows. It states that the line integral of \(\vec{H}\) around any closed loop equals the total current threading that loop:

Applying Stokes' theorem (Chapter 3) to the left side converts it to the point form — Ampère's law as a local statement, and the magnetostatic member of Maxwell's equations:

The cleanest application: an infinite straight wire carrying current \(I\). By symmetry \(\vec{H}\) circles the wire with constant magnitude on any circle of radius \(\rho\). An Amperian circle of circumference \(2\pi\rho\) gives the field in one line:

Notice the \(1/\rho\) falloff — identical in form to the infinite line charge of Chapter 4, but now the field circles rather than radiates. For a coaxial cable, nested Amperian circles give a field that rises inside the inner conductor, falls as \(1/\rho\) between conductors, and vanishes outside — the magnetic version of coaxial shielding.

Winding wire into coils concentrates the field. Inside a long solenoid of \(n\) turns per metre, the field is uniform and axial; outside it is nearly zero. A rectangular Amperian loop gives:

Bending the solenoid into a doughnut makes a toroid of \(N\) total turns, which traps the field entirely inside the core. A circular Amperian loop of radius \(\rho\) through the core gives:

Solenoids and toroids are the workhorses of inductors and transformers — the field-concentrating structures whose stored magnetic energy we quantify in Chapter 14, and which sit at the core of the pulsed-power magnetics in real high-voltage systems.

Magnetic flux \(\Psi_m\) is the amount of \(\vec{B}\) crossing a surface — the magnetic counterpart of electric flux, measured in webers:

Here the mirror with electrostatics finally breaks, in the most important way. There are no magnetic charges — no isolated north or south poles — so magnetic field lines never start or stop; they always close into loops. The net flux out of any closed surface is therefore exactly zero. This is Gauss's law for magnetism, Maxwell's second equation:

For each item, decide whether Ampère's law (high symmetry) or Biot–Savart (finite geometry) is the right tool, then apply it. Difficulty rises down the list.