How Do You Calculate the Flux of a Vector Field Through a Hemisphere?

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SUMMARY

The discussion focuses on calculating the flux of the vector field F, defined by the vector potential A = , through the upper hemisphere of the sphere defined by x^2 + y^2 + z^2 = 1, z ≥ 0. Participants confirm that the flux through the upper hemisphere is zero by applying the divergence theorem, which states that the flux through a closed surface is equal to the volume integral of the divergence of F. Since the divergence of F is zero, the total flux through the closed surface is zero, leading to the conclusion that the flux through the upper hemisphere must also be zero.

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  • Understanding of vector calculus, specifically divergence and curl
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  • #31
Try calculating the flux through the bottom of the hemisphere. It should be obvious to you and your friends when you calculate ##\vec{F}\cdot{\hat{n}}## that the flux through that surface is 0. Since that surface is bounded by the same contour, Stokes' theorem tells you it's equal to ##\iint \vec{A}\cdot d\vec{r}## as well.

Alternatively, by the divergence theorem, you know that the sum of the flux through the bottom and the flux through the top is equal to the volume integral of the divergence, which you found was 0. Therefore, the flux through the top must also vanish.
 

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