# Proof for $W_{\text{total}}$

Looking upon a certain solution of a problem in Griffiths regarding the work done in arranging two concentric spheres with inner shell having charge $$q$$ and outer shell with charge $$-q$$ and I found a statement as superposition of work, and an expression as $$W_{\text{total}}=W_{1}+W_2+\epsilon_{o}\int (E_1\cdot E_2 )d\tau$$

Is there any proof for this expression, as I'm not able to find any references for this expression.

• What are $W_1$ and $W_2$? Commented Dec 16, 2018 at 13:10
• $W_{1}=\frac{q^2}{8 \pi \epsilon_{o} a}$ and $W_{2}=\frac{q^2}{8 \pi \epsilon_{o} b}$ , where $a$ is the radius of inner sphere and $b$ is the radius of outer sphere. Commented Dec 16, 2018 at 13:13

If the electric fields due to the two charges are $$\vec E_1(r)$$ and $$\vec E_2(r)$$ then the resultant field due to the two charges is $$E(r) = \vec E_1(r)+\vec E_2(r)$$
The energy density is $$\frac 12\epsilon_0 \vec E(r)^2 = \frac 12\epsilon_0(\vec E_1(r)+\vec E_2(r))^2= \frac 12\epsilon_0(E_1^2(r) + E_2^2(r) + 2 \vec E_1(r)\cdot \vec E_2(r))$$
• How the third term $\epsilon_{o} \int E_1 \cdot E_2 d\tau$ came? Commented Dec 16, 2018 at 13:28