Magnetostatics

Magnetostatics describes the relationship between electrical currents and magnetic fields. Any electrical current generates a magnetic field. If the current is known, the field can be calculated and if the field is known the current can be calculated. The magnetic field generated by constant electrical currents can be calculated using the Biot-Savart law,

\begin{equation} \vec{B}(\vec{r})=\frac{\mu_0}{4\pi}\int\frac{I d\vec{r}_{wire} \times (\vec{r}-\vec{r}_{wire})}{|\vec{r}-\vec{r}_{wire}|^3}\hspace{1cm}\text{[T]}. \end{equation}

Here $\mu_0= 4\pi \times 10^{-7}$ T m/A is the permeability constant. In general, this integral must be done numerically but there are two special cases where the integral can be performed: an infinitely long straight wire and a solenoid.

Once the magnetic field is known, it can be used to calculate the force on a current carrying wire. This force can be calculated by summing the Lorentz force law $\vec{F}=q\vec{v}\times\vec{B}$ for every particle with charge $q$ and velocity $\vec{v}$ that make up the current. This force is exploited in electric motors.

If the magnetic field is known, the current density $\vec{J}$ can be determined by Ampère's law. There are two forms of Ampère's law. The differential form allows us to calculate the current density,

$$\nabla\times\vec{B}=\mu_0 \vec{J}.$$

Here the curl $\nabla\times\vec{B}$ is defined as,

$$\nabla\times\vec{B}=\left(\frac{\partial B_z}{\partial y}-\frac{\partial B_y}{\partial z}\right)\hat{x}+ \left(\frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x}\right)\hat{y}+ \left(\frac{\partial B_y}{\partial x}-\frac{\partial B_x}{\partial y}\right)\hat{z}.$$

Ampère's law can be rewritten in the integral form using Stokes's theorem. The integral form relates the line integral of the magnetic field going once around a closed curve $C$ to the current $I_{enc}$ passing through the curve,

$$\oint\limits_{C}\vec{B}\cdot d\vec{l}=\mu_0 I_{enc}.$$

Ampère's law can be illustrated with the following simulation.


http://public.mitx.mit.edu/gwt-teal/AmperesLaw.html