If
\[\mathbf{H}\]
be the magnetic field intensity due a current density \[\mathbf{j}\]
in a region of space.\[S\]
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\[\oint_C \mathbf{H} \cdot d \mathbf{r} = \int \int_S (\mathbf{\nabla} \times \mathbf{H}) \cdot \mathbf{n} dS = \int \int_S \mathbf{j} \cdot \mathbf{n} = I\]