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?

  • $\begingroup$ why do you think E1*E2 is something meaningful ? and what the meaning of the Integral ? $\endgroup$ – trula Sep 27 '19 at 20:34
  • $\begingroup$ I dont think its meaningful. Its just what the question asked. $\endgroup$ – Sam Sep 27 '19 at 20:35
  • $\begingroup$ 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. $\endgroup$ – G. Smith Sep 28 '19 at 2:24

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