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Fabio010
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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??