## Proof That Integral of Magnetic Field Strength Around a Closed Loop Equals Current Through the Loop

Theorem
If
$\mathbf{H}$
be the magnetic field intensity due a current density
$\mathbf{j}$
in a region of space. We can draw any surface
$S$
in the space, with boundary
$C$
.
Then
$\oint \mathbf{H} \cdot d \mathbf{r} = I$

Proof
Use Stoke's Theorem
$\oint_C \mathbf{F} \cdot d \mathbf{r} = \int \int_S (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{n} dS$

One of Maxwell's Laws states
$\mathbf{\nabla} \times \mathbf{H} = I$
where
$H$
is the magnetic field strength and
$I$
is the magnetic field strength, so let
$\mathbf{F} = \mathbf{H}$
to give
$\oint_C \mathbf{H} \cdot d \mathbf{r} = \int \int_S (\mathbf{\nabla} \times \mathbf{H}) \cdot \mathbf{n} dS = \int \int_S \mathbf{j} \cdot \mathbf{n} = I$