Activities
Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.
Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .
Students set up and compute a scalar surface integral.
For some integrals, you may wish to use the fact that \[\cos(2\alpha)=2\cos^2\!\alpha-1=1-2\sin^2\!\alpha\]
- A right circular cone has circular base of radius R and height H, both measured in feet.
- What is the volume of the cone?
- Write down as many different integrals as you can for computing this volume.
- Do at least two of these integrals.
Problem
- Find \(dA\) on the surface of an (open) cone in both cylindrical and spherical coordinates. Hint: Be smart about how you coordinatize the cone.
- Using integration, find the surface area of an (open) cone with height \(H\) and radius \(R\). Do this problem in both cylindrical and spherical coordinates.