A spherical volume of radius $a$ is filled with charge density $\rho$. What is the potential energy $U$ of this sphere (what is the work done in assembling it)?

I can't really wrap my mind around this seemingly simple question - I basically used the work equation for a continuous charge distribution, $$W=\dfrac{\epsilon_0}{2}\displaystyle\int_{}{E^2}\hspace{1mm}d\tau$$ (integrating over all space), and got an energy of $$U=\dfrac{Q^2}{8\pi \epsilon_0 a}$$ where $Q$ is the charge enclosed. The correct answer given is instead $$W=\dfrac{3Q^2}{20\pi \epsilon_0 a}$$ Am I using the correct procedure to solve the problem?

  • 2
    $\begingroup$ Did you integrate inside the sphere too? $\endgroup$
    – mmesser314
    Mar 7, 2021 at 16:46
  • $\begingroup$ Have ya tried using Gauss Law ? $\endgroup$ Mar 7, 2021 at 17:05

1 Answer 1


The second equation is the correct equation for a charged spherical volume (see https://www.feynmanlectures.caltech.edu/II_08.html ). The equation you derived is for a charged spherical shell (see http://www.atmo.arizona.edu/students/courselinks/spring13/atmo589/ATMO489_online/lecture_10/potential_energy/charged_sphere_pot_energy.html )

  • $\begingroup$ the link helped a lot, I think I got it - thank you! $\endgroup$ Mar 7, 2021 at 22:59

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