## Derivation of Oerstead's Law from Maxwell's Laws

Stoke's Theorem states
$\oint_C \mathbf{E} \cdot d \mathbf{r} = \int \int_S (\mathbf{\nabla} \times \mathbf{E}) \cdot \mathbf{n} dS$
for a vector field
$\mathbf{E}$
with continuous derivatives.
Let
$\mathbf{F}$
be the magnetic field strength
$\mathbf{H}$
and use Maxwell's Law
$\mathbf{\nabla} \times \mathbf{H} = \mathbf{j} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$
to get
$\oint_C \mathbf{H} \cdot d \mathbf{r} = \int \int_S (\mathbf{j} + \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}) \cdot \mathbf{n} dS$

If the electric field is steady all time derivatives are zero, and we obtain Oersted's Law:
$\oint_C \mathbf{H} \cdot d \mathbf{r} = \int \int_S \mathbf{j} \cdot \mathbf{n} dS$