Ampère's Force Law

This proof is about Ampère's Force Law. For other uses, see Ampère's Law.

Theorem

Let $s_1$ and $s_2$ be wires in a vacuum carrying steady currents $I_1$ and $I_2$.

Then the force between $s_1$ and $s_2$ is given by:

$\ds \mathbf F \propto I_1 I_2 \oint_{s_1} \oint_{s_2} \rd \mathbf l_1 \times \paren {\dfrac {\d \mathbf l_2 \times \paren {\mathbf r_1 - \mathbf r_2} } {\size {\mathbf r_1 - \mathbf r_2}^3} }$

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 $\d \mathbf l_1$.

SI Units

In SI units, Ampère's Force Law is expressed as:

$\ds \mathbf F = \dfrac {\mu_0} {4 \pi} I_1 I_2 \oint_{s_1} \oint_{s_2} \rd \mathbf l_1 \times \paren {\dfrac {\d \mathbf l_2 \times \paren {\mathbf r_1 - \mathbf r_2} } {\size {\mathbf r_1 - \mathbf r_2}^3} }$

where:

$\mathbf F$ is expressed in newtons

$\mathbf r_1 - \mathbf r_2$ is expressed in metres

$I_1$ and $I_2$ are expressed in amperes

$\mu_0$ denotes the vacuum permeability whose value is given by:

$\mu_0 = 1 \cdotp 25663 \, 70621 \, 2 (19) \times 10^{-6}$ henries per metre.

Unrationalized CGS Units

Ampère's Force Law/Unrationalized CGS Units

Proof

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Also see

• Ampère-Maxwell Law‎

Source of Name

This entry was named for André-Marie Ampère.

Sources

• 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Appendix $\text A$: Units