Ampere's equation

In physics, more particularly in electrodynamics, Ampère's equation describes the force between two current-carrying wires. We will list two common cases in two common systems of electromagnetic units. We define the unit dependent constant k as follows,

$k={\begin{cases}{\displaystyle {\frac {\mu _{0}\mu _{r}}{4\pi }}}&{\hbox{for SI units}}\\{\displaystyle {\frac {1}{c^{2}}}}&{\hbox{for Gaussian (rationalized cgs) units}}\end{cases}}$

Here μ₀ is the magnetic permeability of the vacuum and μ_r is the relative magnetic permeability. The quantity c is the velocity of light in the vacuum.

Two straight infinite and parallel wires

The force per unit of length between two straight parallel wires a distance r apart is,

$F=2k{\frac {i_{1}i_{2}}{r}}.\,$

The force is attractive if the currents run in the same direction and repulsive if they flow in opposite direction.

Two loops

Illustration to force between two closed current-carrying loops

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Let $\scriptstyle i_{1}$ and $\scriptstyle i_{2}$ be electric currents constant in time. They run in separate loops, see figure on the right, where all quantities are defined. The total force between two loops is given by the double path integral over the loops

$\mathbf {F} _{12}=-ki_{1}i_{2}\oint _{l_{1}}\oint _{l_{2}}{\frac {(d\mathbf {l} _{1}\cdot d\mathbf {l} _{2})\,\mathbf {r} _{12}}{|\mathbf {r} _{12}|^{3}}},$