Biot-Savart Law

Theorem
Let $s$ be a wire carrying a steady current $I$.

Let $\map {\mathbf B} {\mathbf r}$ be the total magnetic field due to $I$ flowing through $s$ at a point $P$ whose position vector is $\mathbf r$.

Then:
 * $\ds \map {\mathbf B} {\mathbf r} = \dfrac {\mu_0 I} {4 \pi} \oint_s \dfrac {\d \mathbf l \times \paren {\mathbf r - \mathbf r'} } {\size {\mathbf r - \mathbf r'}^3}$

where:
 * $\d \mathbf l$ is an infinitesimal vector associated with $s$
 * $\mathbf r$ is the position vector of $P$
 * $\mathbf r'$ is the position vector of $\d \mathbf l$
 * $\mu_0$ denotes the vacuum permeability.