Application of Gauss Law?

Question

I am applying Gauss' Law to an elementary situation: A sphere of radius r containing a uniform charge density rho, with an electric field normal to the surface of constant value E. Gauss' Law has two forms, integral and differential. See http://en.wikipedia.org/wiki/Gauss%27_law In the integral form, the surface integral of E (normal component) equals the total charge enclosed divided by e0. Since E is constant, the surface integral result is just E times the surface area 4*pi*r^2*E. The total charge enclosed is rho times the spherical volume, or (4/3)*pi*r^3*rho; therefore 4*pi*r^2*E = (4/3)*pi*r^3*rho/e0. The result is E=(1/3)*r*rho/e0.

The other form is div(D)=rho, or div(E)=rho/e0. In spherical coordinates div(E)=(1/r^2)d/dr(r^2*E). For E=constant, this is (1/r^2)*(E*2*r), or 2*E/r. So using the differential form I get
E=(1/2)*r*rho/e0.

One method gives a 1/3 multiplier, the other a 1/2 multiplier. What am I doing wrong? Which one is right?

I thought that divergence was a first derivative function. That is what is implied here http://en.wikipedia.org/wiki/Nabla_in_cylindrical_and_spherical_coordinates

Answer 1

OK your first error is in the div(E)
for spherical coordinates div(E)=1/r^2d/dr(r^2 dE/dr)

I would stick with the integral form, because you don't know E as a function of r. You just know its value on the surface. But you can infer a usable equation for E(r) if you really want to.