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## Homework Statement

Find an expression for the electrostatic self-energy of an arbitrary spherically symmetric charge density distribution ρ(r). You may not assume that ρ(r) represents any point charge, or that it is constant, or that it is piecewise constant, or that it does or does not cut off at any finite radius r. Your expression must cover all possibilities. Your expression may include an integral or integrals which cannot be evaluated without knowing the specific form of (r).

## Homework Equations

[itex]\phi[/itex] = U/q

F=-∇U

E=-∇[itex]\phi[/itex]

U = (1/2) [itex]\int[/itex] ρ[itex]\phi[/itex]d

^{3}r

## The Attempt at a Solution

For an arbitrary continuous charge distribution:

[itex]\phi[/itex](r) = [itex]\frac{1}{4*pi*ε(0)}[/itex] [itex]\int[/itex][itex]\frac{ρ(r')}{abs(r - r')}[/itex]d

^{3}r'

For spherically-symmetric distribution of charge:

[itex]\phi[/itex] = ([itex]\frac{1}{4*pi*ε(0)}[/itex] 2[itex]\pi[/itex] ∫ sin[itex]\theta[/itex]'d[itex]\theta[/itex]' ∫ r'

^{2}dr' [itex]\frac{ρ(r')}{r''}[/itex] cos[itex]\theta[/itex])

Then solving for U:

U = (1/2) ∫ρ ([itex]\frac{1}{4*pi*ε(0)}[/itex] 2[itex]\pi[/itex] ∫ sin[itex]\theta[/itex]'d[itex]\theta[/itex]' ∫r'

^{2}dr' [itex]\frac{ρ(r')}{r''}[/itex] cos[itex]\theta[/itex]) d

^{3}r

Could someone please tell me if I made any errors or assumptions that I should not have made for this situation?

Note: I couldn't figure out how to put limits on the integrals, but for the spherically-symmetric distribution of charge first integral (sin[itex]\theta[/itex]'d[itex]\theta[/itex]') is from 0 to pi and the second integral (r'

^{2}dr' [itex]\frac{ρ(r')}{r''}[/itex] cos[itex]\theta[/itex]) is from 0 to infinity. These also affect the equation for U, and there is no limits on the first integral in U.