Magnetic Fields and Electromagnetism

A magnetic field fills the space around magnets and currents and pushes on moving charges, currents and magnetic materials. Magnets always come as dipoles with a north and a south pole, and field lines form closed loops, which is Gauss’s law for magnetism: the net magnetic flux through any closed surface is zero. Ferromagnets like iron can be permanently magnetized, paramagnets respond weakly, every material shows a weak opposing diamagnetism, and permeability measures how strongly a material magnetizes.

A magnetic field pushes on a moving charge with $\vec{F}_B = q\vec{v} \times \vec{B}$, a force perpendicular to both the velocity and the field that you find with the right-hand rule. Because the force is always sideways, a charge moving perpendicular to a uniform field travels in a circle. A moving charge also makes its own magnetic field, and charges pushed sideways inside a current-carrying conductor create a small voltage across it, called the Hall effect.

The Biot–Savart law, $d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{\ell} \times \hat{r}}{r^2}$, gives the field from each small piece of a current, and you add the pieces with an integral. It shows that field lines circle a wire and gives results like $B = \frac{\mu_0 I}{2R}$ at the center of a circular loop. A magnetic field also pushes on a current-carrying wire, $\vec{F} = I\vec{\ell} \times \vec{B}$.

Ampère’s law, $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{enc}$, links the magnetic field around a closed Amperian loop to the current passing through it. For symmetric cases it quickly gives the field of a long straight wire, $B = \frac{\mu_0 I}{2\pi r}$, and inside a long solenoid, $B = \mu_0 n I$. Maxwell added that a changing electric field also makes a magnetic field.