Question

# Given that B = curl A, use the divergence theorem to show that $\oint \mathbf{B} \cdot \mathbf{n} d \sigma$ over any closed surface is zero.

Solution

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Using the divergence theorem: $\oint_{\partial v} \boldsymbol{B}\cdot\boldsymbol{n}d\sigma=\int_{v}(\boldsymbol{\nabla}\cdot\boldsymbol{B})d\tau$, but $\boldsymbol{B}=\boldsymbol{\nabla}\times \boldsymbol{A}$, then:

$\oint_{\partial v} \boldsymbol{B}\cdot\boldsymbol{n}d\sigma=\int_{v}\boldsymbol{\nabla}\cdot(\boldsymbol{\nabla}\times \boldsymbol{A})d\tau$

but $\boldsymbol{\nabla}\cdot(\boldsymbol{\nabla}\times \boldsymbol{A})=0$ everywhere, so:

$\oint_{\partial v} \boldsymbol{B}\cdot\boldsymbol{n}d\sigma=0$