Ampère's Force Law

Theorem
Let $s_1$ and $s_2$ be wires carrying steady currents $I_1$ and $I_2$.

Then the force between $s_1$ and $s_2$ is given by:
 * $\ds \mathbf F = \dfrac {\mu_0} {4 \pi} I_1 I_2 \oint_{s_1} \oint_{s_2} \rd \mathbf l_1 \times \paren {\dfrac {\d \mathbf l_2 \times \paren {\mathbf r_1 - \mathbf r_2} } {\size {\mathbf r_1 - \mathbf r_2}^3} }$

where:
 * $\d \mathbf l_1$ and $\d \mathbf l_2$ are infinitesimal vectors associated with $s_1$ and $s_2$ respectively
 * $\mathbf r_1$ and $\mathbf r_2$ are the position vectors pointing from $\d \mathbf l_2$ towards $\d \mathbf l_1$
 * $\mu_0$ denotes the vacuum permeability.