Electrostatic energy of spherical shell.

Homework Statement

Determine the electrostatic energy, W, of a spherical shell of radius R with total charge q, uniformly distributed. Compute it with the following methods:

a) Calculate the potential V in spherical shell and calculate the energy with the equation:

W = (1/2) * ∫σVda

b) Calculate the electric field inside and outside, and calculate the energy with the equation:

W = (1/2) *εo ∫ E^2 dV

The Attempt at a Solution

a)
Electric filed inside = 0

Electric field outside = (1/4piεo) * Q/r^2

The Potential is v(r)-v(infinity) = - ∫(infinity to R) (Electric field outside) dr

= (1/4piεo) * Q/R

W = (1/2) * ∫σVda

because dq = σda

W = (1/2) * ∫(1/4piεo) * q/R * dq = (1/8piεo) * Q^2/R

b) The electric field is already calculated...

W = (1/2) *εo ∫ E^2 dV

= (1/2) *εo ∫ (1/16pi^2εo^2) * q^2/r^4 dV

(1/2) *εo ∫(0 to 2pi) ∫(0 to pi) ∫ (0 to R ) [(1/16pi^2εo^2) * q^2/r^4 ] * r^2 sinθ dr dθ dβ

= -(1/8piεo) * Q^2/R

The problem is, the difference in the signal...isn't it supposed to be equal??

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W = (1/2) * ∫σVda

because dq = σda

W = (1/2) * ∫(1/4piεo) * q/R * dq = (1/8piεo) * Q^2/R
You undid the simplification that was made for you :)
W = 1/2 ∫ V dq = 1/2 ∫ σV da

The q inside your integral is the constant Q, not little q that varies, but I think you made a typo since the constants are still right.

b) The electric field is already calculated...

W = (1/2) *εo ∫ E^2 dV

= (1/2) *εo ∫ (1/16pi^2εo^2) * q^2/r^4 dV

(1/2) *εo ∫(0 to 2pi) ∫(0 to pi) ∫ (0 to R ) [(1/16pi^2εo^2) * q^2/r^4 ] * r^2 sinθ dr dθ dβ
Didn't you say in part a that the E field inside the sphere was everywhere 0?

You undid the simplification that was made for you :)
W = 1/2 ∫ V dq = 1/2 ∫ σV da

The q inside your integral is the constant Q, not little q that varies, but I think you made a typo since the constants are still right.

Didn't you say in part a that the E field inside the sphere was everywhere 0?

So the energy is zero?

the integration must be from R to infinity,not from 0 to R.

the integration must be from R to infinity,not from 0 to R.
yeh, it makes sense. And the result is positive.

Thanks for the help.