How Do You Calculate the Electrostatic Energy of a Hollow Conducting Sphere?

[sal1234]

[Note from mentor: this was originally posted in a non-homework forum, so it does not use the homework template.]

There is a general relation between the work U required to assemble a charge distribution ρ and the potential φ(r) of that distribution:
U = 1/2 ∫ ρ φ dv
Now using this specific formula i would like to calculate the electrostatic energy of a Hollow conducting Sphere of Radius ''R'' which is given a Charge ''Q''.

MY ATTEMPT AT THE SOLUTION:

WKT, φ(r) = (1/4πε0)Q/r For every r > R

ρ = Q/4πR^2

Now let us consider a spherical element of radius "r" and thickness "dr". The potential is uniform in this small element.The energy in this small element is
du = (1/2 )ρ φ dv

dv = area * thickness
dv = 4πr^2 * dr

du = (1/2 ) ρ (1/4πε0) Q/r dr

integerating from R to infinity in order to get the total energy

U = ∫(1/2 ) ρ (1/4πε0) Q/r dr

But i am not able to get right answer.

 

[sal1234]

the image link

 

[Arman777]

integerating from R to infinity in order to get the total energy

Maybe you made something wrong here or later parts. Its hard for me tell, but it can be sometimes hard to take the integral. Try to write the $dV$ in terms of $dR$ and then $dR$ in terms of $dQ$ and then try to take integral. By doing this you ll reduce your calculations a lot. Also make a strict destinction between $R$ and $r$. And open the $ρ$ When you are taking the integral. It should come out right then..

 

[vela]

You need to rethink your integration. The charge density is zero for $r \ne R$.

 

[sal1234]

You need to rethink your integration. The charge density is zero for $r \ne R$.

what should i do?

 

[Delta2]

You need to carefully define the volume charge density (the surface charge density is as you say, however you need the volume charge density since you going to integrate over a volume) for your problem with the help of a dirac delta function. Then the integral should be easy to calculate.