Magnetic Field and Ampere's Law

Magnetic Field Basics

Magnetism is electricity's velocity-dependent twin: it acts only on moving charges, always at right angles to their motion.

The magnetic force on a charge is given by $\vec{F} = q\vec{v}\times\vec{B}$, where $q$ is charge, $\vec{v}$ is velocity, and $\vec{B}$ is the magnetic field.

Because this force is always perpendicular to $\vec{v}$, it changes direction but never speed or kinetic energy. If a problem needs energy added, an electric field must do it.

The Biot-Savart law gives the field from a current element:

$d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{l}\times\hat{r}}{r^2}$

Here $\mu_0$ is the permeability of free space, $I$ is current, $d\vec{l}$ is length, and $r$ is distance.

Ampere's Law and Results

Ampere's law relates magnetic fields to steady currents with symmetry:

$\oint_C \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}}$

The force between parallel wires carrying currents $I_1$ and $I_2$ is $F/L = \dfrac{\mu_0 I_1 I_2}{2\pi d}$. Parallel currents attract; antiparallel repel.

Physics link: A charge in a uniform $\vec{B}$ moves in a circle of radius $r = mv/(qB)$, used in cyclotrons.