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## 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??