(Electrostatics) Energy of a configuration

LuisVela

1. The problem statement, all variables and given/known data

For a given configuration i found the scalar potential [tex]\phi(r)[/tex]-->(as you can see its a function only of r)
My question is about calculating the energy of the system.


2. Relevant equations

[tex]
W=-\dfrac{\varepsilon_0}{2}\int |\nabla \phi|^2 d^{3}x =\dfrac{1}{2}\int \phi \rho \,d^{3}x
[/tex]


3. The attempt at a solution

I just dont know if i should integrate [tex]\phi (r)[/tex] like a triple integral with limits [tex](0,\infty)x(0,2\pi)x(0,\pi)[/tex] or should i perform the inverse substitution (from sferical coordinates to cartesian ) and then integrate [tex]\phi (x,y,z)[/tex] like a triple integral with limits (-oo,oo)x(-oo,oo)x(-oo,oo)

Moreover, if i perform the change in the variables,what happen to the Jacobian of the substitution (|J|=[tex]r^2 sin(\vartheta)[/tex] )????