## Proof That the Magnetic Flux Through a Closed Surface is Zero

Theorem
The total magnetic flux through a closed surface is zero. Proof
The total magnetic flux through a closed surface is
$\Phi = \int \int_S \mathbf{B} \cdot \mathbf{n} dS$
.
We can use the Divergence Theorem to equate this to a volume integral.
$\int \int_S \mathbf{B} \cdot \mathbf{n} dS = \int \int \int_V \mathbf{\nabla} \cdot \mathbf{B} dV$
.
$V$
in this integral is the volume enclosed by
$S$
. But
$\mathbf{\nabla} \cdot \mathbf{B} =0$
from Maxwell's equations so
$\Phi = \int \int_S \mathbf{B} \cdot \mathbf{n} dS =0$
.