(Electrostatics) Energy of a configuration

[LuisVela]

Homework Statement

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.

Homework Equations

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

The Attempt at a Solution

I just don't 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] )?

 

[Jhero]

your response to this forum post would be to provide guidance on how to calculate the energy of the system. You could suggest that the poster use the second equation provided, as it is specifically for calculating the energy of a system using the scalar potential. You could also clarify that the integration should be performed over all space, not just within certain limits. Additionally, you could explain that the Jacobian factor should be included in the integration when performing the change of variables. Finally, you could suggest that the poster double-check their calculations and units to ensure accuracy.