# How to perform integration for all of space?

Two conducting spheres of radii $$a$$ and $$b$$ have charges $$Q_1$$ and $$Q_2$$. Distance between their centres is $$R$$. If $${\vec E_1}$$ and $${\vec E_2}$$ are electric field vectors due to the two spheres at any point, then what is the value of $${\varepsilon_0}{\int_{All\ space}}{\vec E_1}{\cdot}{\vec E_2}\;dV$$

I am a high school student and came acros this question in one of my reference books. I have done electrostatics questions that require integration in the past but none of them had the "All space" in integration. How do I even approach this problem?

• why do you think E1*E2 is something meaningful ? and what the meaning of the Integral ? – trula Sep 27 '19 at 20:34
• I dont think its meaningful. Its just what the question asked. – Sam Sep 27 '19 at 20:35
• This integral arises from integrating the electrostatic energy density, $\frac 12\epsilon_0\mathbf{E}^2$. It’s the “cross term” representing the potential energy of interaction of the two spheres. – G. Smith Sep 28 '19 at 2:24