# Homework Help: Energy of Spherical Distribution

1. Feb 20, 2008

### mgiddy911

1. The problem statement, all variables and given/known data

Determine the energy required to generate a spherical charge distribution
of radius R and uniform density ρ0 . In class, we went over two different
methods. For this assignment, you need to work out the following two
integrals over a volume (and surface) of a concentric sphere of radius a,
where a > R.

U = $$\epsilon//2$$ [ $$\int$$E^2 +$$\oint$$ VE ]
here the first integral is a volume integral over the sphere and the second integral is a surface integral over the surface of the sphere.
this is equation 2.44 in Griffiths

2. Relevant equations
I started with the equation for electric field inside a sphere of uniform distribution

which is E = kQr/R^3 in radially outward, where k is the 1/4 pi epsilon constant and r is some some distance from the center less than R

Then I use the equation for Potential inside the sphere as V= (kQ/2R)(3-r^2/R) again where r is less than R

3. The attempt at a solution

I am just stuck with the integration I guess. The first integral uses the square of E which is easily found by just squaring the equation above, so that now E^2 is not a vector but just the scalar square of the equation above. then dTau for the volume of the sphere is just the volume element for spherical coordinates? (r^2)sin(phi)drd(phi)d(theta)

The second integral uses the V equation, multiplied by the E equation and integrates over the surface where the da = area element for area of surface of sphere = (r^2)sin(phi)d(phi)d(theta)

I apologize if this is hard to read, I am terrrrrrible at using the latex stuff, it ever seems to work out

Edit: I don't yet have a complete solution but I just realized my confusion was coming form mistranslation in the coordinate systems. I always forget what theta and phi correspond to as i have had teachers switch them up before. I saw the sin(phi) thinking the integrals were going to integrate sin (phi) from 0---2Pi which would yield zero

Last edited: Feb 20, 2008