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## Homework Statement

The space between the two concentric spheres is charged by spatial density of charge [tex]\rho=\frac{\alpha}{r^2}[/tex]. The radius of spheres are [tex]R_1,R_2[/tex]. Integral charge is [tex]Q[/tex]. Find energy of electrostatic field.

## Homework Equations

Gauss law

[tex]\oint_S\vec{E} \cdot d{\vec{S}}=\frac{q}{\epsilon_0}[/tex]

[tex]W_E=\frac{1}{2}\epsilon_0\int_VE^24\pi r^2dr[/tex]

## The Attempt at a Solution

Using Gauss law I get

[tex]E^{(1)}=0[/tex], for [tex]r<R_1[/tex]

[tex]E^{(2)}(r)=\frac{1}{\epsilon_0}\cdot \frac{Q}{4\pi(R_2-R_1)}\frac{r-R_1}{r^2}[/tex]

for [tex]R_1\leq r \leq R_2[/tex]

[tex]E^{(3)}(r)=\frac{Q}{4\pi\epsilon_0r^2}[/tex], for [tex]r>R_2[/tex]

And get [tex]W_E^{(1)}=0[/tex]

[tex]W_E^{(2)}=\frac{Q^2}{8\pi\epsilon_0(R_2-R_1)}(1+\frac{2R_1}{R_2-R_1}ln\frac{R_2}{R_1}+\frac{R_1}{R_2})[/tex]

[tex]W_E^{(3)}=\frac{Q^2}{8\pi\epsilon_0R_2}[/tex]

This is my solution.

Final solution from book is

[tex]W_E^{(1)}=W_E^{(3)}=0[/tex]

[tex]W_E^{(2)}=\frac{Q^2}{4\pi\epsilon_0(R_2-R_1)}(1+\frac{2R_1}{R_2-R_1}ln\frac{R_2}{R_1})[/tex]

Where I make a mistake?