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

Click For Summary

Homework Help Overview

The problem involves calculating the flux of a vector field through the upper hemisphere of a sphere defined by the equation x²+y²+z²=1, with z≥0. The vector field F is given as having a vector potential A = .

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants discuss the relationship between the vector potential A and the vector field F, questioning whether to take the gradient or the curl to find F. There are considerations of using Stokes' theorem and the divergence theorem to calculate the flux.

Discussion Status

The discussion is ongoing, with participants exploring different methods to approach the problem. Some suggest using Stokes' theorem while others consider the implications of the divergence theorem. There is a recognition that the flux through the entire closed surface is zero, but the specific flux through the upper hemisphere remains a point of contention.

Contextual Notes

Participants note that the problem specifies the flux through the upper hemisphere, which may differ from the flux through a closed surface. There is also mention of the need to parameterize the surface and the implications of normal vectors in the context of the hemisphere.

  • #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.
 

Similar threads

Find surface of maximum flux given the vector field's potential
  • · Replies 6 ·
Replies
6
Views
2K
Gauss' Theorem - Net Flux Out - Comparing two vector Fields
  • · Replies 5 ·
Replies
5
Views
1K
Flux of a vector field through a surface
  • · Replies 8 ·
Replies
8
Views
2K
Given a set of solids, compute the inward flux
  • · Replies 3 ·
Replies
3
Views
2K
Can't find potential of vector field
  • · Replies 10 ·
Replies
10
Views
2K
Finding Flux Through a Cylinder with the Divergance Theorom
  • · Replies 1 ·
Replies
1
Views
1K
Compute the flux of a vector field through the boundary of a solid
  • · Replies 2 ·
Replies
2
Views
2K
Lagrange multipliers understanding
  • · Replies 8 ·
Replies
8
Views
2K
Verifying the flux transport theorem
  • · Replies 4 ·
Replies
4
Views
2K
Stokes' Theorem for a Circle in a Plane
  • · Replies 7 ·
Replies
7
Views
2K