Ampère's Law

Disambiguation

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Ampère's Law may refer to:

Ampère's Force Law

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 $\d \mathbf l_1$.

Ampère's Circuital Law, also known as the Ampère-Maxwell Law

Let $\mathbf B$ be a magnetic field due to a steady current $I$ flowing through a wire $s$.

Then:

$\ds \oint \mathbf B \cdot \rd \mathbf l = \mu_0 I$

where:

the line integral is taken around a closed path

$\d \mathbf l$ is an infinitesimal vector associated with $s$

$\mu_0$ denotes the vacuum permeability.

That is, the line integral of $\mathbf B$ through the area enclosed by the closed path is equal to $\mu_0 I$.

Ampère's Law with Maxwell's Addition

$\nabla \times \mathbf B = \mu_0 \paren {\mathbf J + \varepsilon_0 \dfrac {\partial \mathbf E} {\partial t} } $

Source of Name

This entry was named for André-Marie Ampère.