Electrostatic Energy of a solid sphere with a cavity

[Arman777]

Homework Statement

Determine the electrostatic energy U stored in the sphere

Relevant Equations

$W = ε_0/2 \int E^2d\tau$ for all space

I tried to use $W = ε_0/2 \int E^2d\tau$ for all space. So I find that $E = \frac{(R^3 - b^3)\rho}{3ε_0r^2}$ where $\rho$ is the charge denisty. So from here when I plug the equation I get something like

$$W = \frac{(R^3 - b^3)^2\rho^2 4 \ pi}{18ε_0} \int_{?}^{\inf}1/r^2dr$$

Is this approach correct ?

At this point I get stuck for the boundary conditions. If I put R I get something meaningless

 

[TSny]

I tried to use $W = ε_0/2 \int E^2d\tau$ for all space.

OK

So I find that $E = \frac{(R^3 - b^3)\rho}{3ε_0r^2}$ where $\rho$ is the charge denisty.

For what region of space is this expression for E applicable? You have three distinct regions to consider.

 

[Arman777]

OK
For what region of space is this expression for E applicable? You have three distinct regions to consider.

This is for $r>R$ so I have to also find $E(r<b)$, $E(b<r<R)$ ?

Then The U will be,

$$U = \frac{(R^3 - b^3)^2\rho^2 4 \ pi}{18ε_0} \int_{R}^{\inf}1/r^2dr + \frac{e_0}{2} \int_{b}^{R}\frac{\rho (r - b^3)}{3ε_0}dr$$

Where $E = \frac{\rho (r - b^3)}{3ε_0}, (b<r<R)$

 

[Delta2]

I think the correct expression for the E-field for $b<r<R$ is $$E=\frac{\rho(r^3-b^3)}{3\epsilon_0r^2}$$.