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.
assignment Homework
assignment Homework
Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).
keyboard Computational Activity
120 min.
assignment Homework
assignment Homework
Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.
group Small Group Activity
30 min.
Students use known algebraic expressions for vector line elements \(d\vec{r}\) to determine all simple vector area \(d\vec{A}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.
This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.
assignment Homework
Find the upward pointing flux of the electric field \(\vec E =E_0\, z\, \hat z\) through the part of the surface \(z=-3 s^2 +12\) (cylindrical coordinates) that sits above the \((x, y)\)--plane.
assignment Homework
group Small Group Activity
5 min.
computer Computer Simulation
30 min.