Deriving Coulomb's Law from Gauss' Law

[Ralru]

Homework Statement

When I try working out the example below from PlanetPhysics, I wind up with 2PI rather than 4PI in my answer. Should I be considering my result valid for only a hemisphere and double it for a sphere--or am I just making a mistake in my math?

"As an example of the statement that Maxwell's equations completely define electromagnetic phenomena, it will be shown that Coulomb's law may be derived from Gauss' Law for electrostatics. Consider a point charge. We can obtain an expression for the Electric Field surrounding the charge. We surround the charge with a "virtual" sphere of radius R, then use Gauss' law in integral form."

"We rewrite this as a volume integral in spherical polar coordinates over the "virtual" sphere mentioned above, which has the point charge at its centre. Since the electric field is spherically symmetric (by assumption) the electric field is constant over this volume" (http://planetphysics.org/encyclopedia/DerivationOfCoulombsLawFromGaussLaw.html).

Homework Equations

[URL]http://www.fortunesofwar.com/C1.bmp[/URL]
[URL]http://www.fortunesofwar.com/C2.bmp[/URL]
[URL]http://www.fortunesofwar.com/C3.bmp[/URL]

The Attempt at a Solution

I first integrate Er sin theta dtheta between the limits of 0 and PI which gives Er(-cos theta) =2Er
Then I integrate 2Er dphi between the limits of 0 and 2PI giving 4'PI'Er
And then I integrate 4'PI'Er dr between the limits of 0 and R, giving 1/2 4'PI'Er^2 evaluated between 0 and R, giving 2'PI'ER^2.

I must be missing something here (it's been a while), so I'd appreciate knowing where I am making my mistake.

Thanks

 

[vela]

You're not making a math mistake. The example is wrong. The electric field isn't constant over the volume, and the integral shouldn't be over the volume of the sphere. The integral should be over the surface area, where the electric field is of constant magnitude.

 

[Ralru]

Thank you very much. It's nice to know that I have not gone completely insane yet. It's been quite a while since I played with this stuff--and after a long hiatus, one begins to doubt his/her own ability. Thanks again.