The Biot-Savart Law

Steady Currents

Every current, steady or not, must obey conservation of charge. Its local form is the continuity equation:

\frac{\partial \rho}{\partial t} + \nabla \cdot \vec{J} = 0

This is the general law: if more current leaves a small region than enters it, the charge density $\rho$ in that region must decrease. A steady current is the special case in which charge flows without piling up anywhere, so $\partial \rho/\partial t=0$. The continuity equation then reduces to:

\nabla \cdot \vec{J} = 0

Thus $\nabla\cdot\vec{J}=0$ is not a separate conservation law; it is the steady-current limit of the full continuity equation. Whatever current enters a volume must then leave it.

Deep Dive: Charge Conservation, Steady Current, and KCL

The Biot-Savart Law

The Biot-Savart Law relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. Consider a steady current $I$ flowing along a path. The magnetic field $\vec{B}$ at a field point $\vec{r}$ generated by a current element at a source point $\vec{r}'$ is given by:

\vec{B}(\vec{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\vec{l}' \times (\vec{r} - \vec{r}')}{|\vec{r} - \vec{r}'|^3} = \frac{\mu_0}{4\pi} I\int \frac{ d\vec{l}' \times \hatrsep}{\rsep^2}

Where:

$\mu_0 = 4\pi \times 10^{-7} \, \text{N/A}^2$ is the permeability of free space.
$d\vec{l}'$ is a vector along the path of the current element.
$\vec{r} - \vec{r}'$ (often denoted as $\vecrsep$) is the separation vector from the source point $\vec{r}'$ to the field point $\vec{r}$.
$\hatrsep$ is the unit displacement vector in that direction.

The magnetic field is measured in Teslas (T), where: \[ 1 \text{ T} = 1 \text{ N}/(\text{A} \cdot \text{m}) \]

Deep Dive: Applications of the Biot-Savart Law