Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .
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Flux through a ConeFind the flux through a cone of height \(H\) and radius \(R\) due to the vector field \(\vec{F} = C\,z\,\hat{z}\).
This activity is part of a sequence on flux Flux Sequence, which we strongly recommend. If you don't have time, the minimum introduction is a short lecture introducing the concept of flux (as the amount of a vector field perpendicular to a surface) and how to calculate it: \[ \Phi = \int_S\, \vec{F}\, \cdot \,d\vec{A}\]
Prompt: Find the flux through a cone of height \(H\) and radius \(R\) due to the vector field \(\vec{F} = C\,z\,\hat{z}\).
This prompt is open-ended in that it doesn't specify either the location of the cone or whether or not the circular top of the cone is to be considered part of the surface. We like to leave it open-ended, see what students do, and when students question the open-endedness, give a mini-”sermon” on the ill-posedness of most real world problems. If you are short of time, or otherwise want to avoid these questions, you should use a more explicit prompt.
If you choose the point of the cone at the origin (and allow it to open upward, like an icecream cone), then the problem can easily be solved in spherical coordinates as well as the obvious cylindrical coordinates.
We do a brief summary of the main points to wrap up the activity.