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

Add comment

Security code
Refresh