What Potential Should Be Used for Energy Stored in a Charged Sphere?

[Kosta1234]

Homework Statement

Energy stored in a charged sphere

Relevant Equations

$$ U = \frac {1}{2} \cdot \int \phi (r) \cdot \rho(r) dV $$

Hi.
When I am asked to figure out the Energy stored in a charged sphere and I use this equation: $U = \frac {1}{2} \cdot \int \phi (r) \cdot \rho(r) dV$
what is the potential $\phi ( r)$ stands for? I tried to use the potential inside the sphere, because out side of the sphere $\rho (r) = 0$, and I tried to sum those to up.
I'm not getting the same answer as in this equation:
$$ U = \frac {\varepsilon }{2} \cdot \int_{all space} E^2 dV $$so what $\phi (r)$ I've to use and why?Edit: I got it right, I think.
was I right when I said to use the potential inside the sphere, because out side of the sphere $\rho (r) = 0$
and to use the $U = \frac {\varepsilon }{2} \cdot \int_{all space} E^2 dV$ to all space?

 

[vanhees71]

It's the electrostatic potential you should use, i.e., the solution of the Legendre equation,
$$\Delta \phi(\vec{x})=-\frac{1}{\epsilon} \rho(\vec{x}).$$
This potential is defined up to a constant, and so is $U$.

It's simple to show that (up to a constant) your two expressions deliver the same result. To see this, just use
$$\vec{E}=-\vec{\nabla} \phi$$
in one of the factors in the integral
$$\tilde{U}=\frac{\epsilon}{2} \int_{\mathbb{R}^3} \mathrm{d}^3 x \vec{E}^2(\vec{x}),$$
and then Gauss's theorem for 3D partial integration! It's a good exercise!

BTW: You can fix the constant by demanding that $U=0$ for $\rho=Q/V$ (where $Q$ is the total charge, and $V$ the volume of the sphere; I guess you mean a homogeneously charged sphere).

You can also find the exact solution of this problem by solving the Legendre equation in shperical coordinates.