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

TheoremThe 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 \[\Phi = \int \int_S \mathbf{B} \cdot \mathbf{n} dS =0\]

.