Question

Use Gauss' Law for Electricity to find the total charge in the cone y=sqrt(x^2+z^2) for E=<2xy, y^2, 5xy>. (Answer: 0) I've set up the triple integral of 4y dV and the formula for volume of a cone is V=1/3*pi*r^2*h but I have trouble finding the radius and the height.

Answer 1

I don't think you can use the equation for volume of a cone straight forward in this application. because$$\iiint 4y \;dV$$ is like finding mass of a cone with the property, ie the deeper you go, higher the density of the cone becomes.

as a matter of fact, I don't even think the answer will be zero

Answer 2

The integration bounds are tricky. The integration over y should be from r cos(45) to H, and the integration over r should be from 0 to Hcos(45).

\[\int_0^{2\pi} \int_0^{H\cos(45)}\int_{r\cos(45)}^H r\cdot 4y \space dy dr d\theta \]