12.1 Magnetic Fields

A magnetic field $\vec{B}$ is a vector field, measured in teslas (T). One tesla is a strong field: a fridge magnet is around 0.005 T near its surface, and Earth's field at the surface is roughly 5 × 10⁻⁵ T. Sources of magnetic fields are moving charges: currents in wires, and the spins and orbits of electrons inside magnets.

Field lines show the direction a compass needle's north end would point at each spot. Outside a magnet they run from the north pole to the south pole; inside the magnet they continue from south back to north. Lines are closer together where the field is stronger, and they never cross.

On a flat drawing, a dot (•) means a field pointing out of the page toward you, like the tip of an arrow. An × means a field pointing into the page, like the tail feathers of an arrow flying away.

Every magnet has a north pole and a south pole; it's a magnetic dipole. Like poles repel and unlike poles attract. Cut a bar magnet in half and you don't get a lone north pole: you get two smaller magnets, each with its own north and south. No one has ever found an isolated magnetic pole, called a monopole.

A current loop is also a magnetic dipole. Curl the fingers of your right hand in the direction of the loop's current, and your thumb points toward its north side.

In a uniform field, a dipole feels no net force, but it does feel a torque that turns it to line up with the field. That's what a compass needle does. In a nonuniform field, a dipole also feels a net force, which is why a magnet attracts a paper clip: the field is stronger closer to the magnet.

A dipole's own field gets weaker as you move away from it, which is why a magnet's pull fades quickly with distance.

Because field lines always form closed loops, any line that enters a closed surface must also leave it. So the net magnetic flux through any closed surface is zero: $\oint \vec{B} \cdot d\vec{A} = 0$.

Compare this with Gauss's law for electric fields (8.6), where the net flux depends on the charge inside. The magnetic version says there's no magnetic charge to enclose. It's one of the four Maxwell's equations.

Earth acts roughly like a giant bar magnet tilted a little from its spin axis. A compass's north end points toward geographic north, so the magnetic pole near the geographic North Pole is actually a south magnetic pole. Away from the equator, Earth's field also points partly into or out of the ground.

Materials respond to a magnetic field in different ways. Permeability $\mu$ tells you how readily a material magnetizes when you put it in a field: the bigger $\mu$, the more the material adds to the field inside it. The permeability of free space is $\mu_0 = 4\pi \times 10^{-7}\text{ T·m/A}$, and it appears in every field formula in this unit.

Don't think of a material's $\mu$ as one number to look up. Heat the material, turn it, or make the applied field stronger or weaker, and $\mu$ can shift, especially for iron and other ferromagnets. Ferromagnetic materials have permeabilities far larger than $\mu_0$; paramagnetic materials are just above it and diamagnetic ones just below.