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The problem:

Consider a uniformly charged sphere of radius $R$ and charge $Q$ and then they separate into two spherical halves of equal volume and charge, and both get to stabilize. Determine the variation of the electrostatic potential energy of the system after the division of the first sphere of fluid into the other two, assuming they are separated by a great distance.

(I'm just ignoring the "fluid" in "sphere of fluid" and assuming it's a typo)

What I think

At the beginning there is only one charge, so the electric potential energy (is it the same as electrostatic potential energy?) is zero. That's because we need at least two charges to talk about potential energy.

After the division into two spheres, we can use the formula: $$ U_f=\frac{1}{4\pi \varepsilon_o} \frac{\frac{Q}{2}\frac{Q}{2}}{L} \approx 0 $$ where $L$ is the distance between the two spherical halves. The electric potential should be zero because the separation is a "great distance". Finally, the change in electrostatic potential is zero, because it's zero at the beginning and it's zero at the end.

Questions

I haven't used the radius $R$ and the problem seems too trivial. I feel like I don't really understand what's going on and am missing something. Also, does the solution differ in any way if we divide the sphere into two halves of a sphere (two semispheres), and not into two spherical halves as the problem says?

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    $\begingroup$ The uniformly charged sphere is not a single charge. There is a potential energy there. $\endgroup$ – user58697 Sep 23 '18 at 22:24
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    $\begingroup$ @evaristegd: I am not sure what is meant by "stabilize", is the shape of the uniformly charged material allowed to change freely and settle into the most stable arrangement, hence the word "fluid"? The difference in electrostatic potential energy is the difference in the work required to assemble the starting and ending configurations. $\endgroup$ – user7777777 Sep 23 '18 at 22:31
  • $\begingroup$ @user7777777 : FWIW, the original version in Spanish says "ambas [mitades esféricas] llegan a estabilizarse.". Your explanation about the "fluid" part makes sense to me too. Also, the problem was part of a sophomore-level E&M test. $\endgroup$ – evaristegd Sep 23 '18 at 23:36
  • $\begingroup$ @user58697 , thank you. I did some googling and found this about the energy of a charged sphere volume. Do you think that source is reliable? I think the problem in this post asks about a spherical shell (sphere). Do you think I can apply the techniques used in the PDF, but for a spherical shell? $\endgroup$ – evaristegd Sep 23 '18 at 23:41
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    $\begingroup$ There are no shells. You start with a spherical drop of some liquid, uniformly charged (volume charge density) and in the end you have two spherical drops of liquid, each half getting half of the charge. You need to know the potential energy for uniformly charged sphere in order to solve the problem. $\endgroup$ – nasu Sep 24 '18 at 1:57
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Initially, we have a uniformly charged sphere with total charge $Q$. In the end, we end up with two uniformly charged spheres, each with half the volume and charge of the original one, separated by an infinite distance.

Since you have already mentioned the charge stored in a uniformly charged sphere is $$\frac{1}{4 \pi \epsilon_0} \frac{3Q^2}{5R}$$ we can compute the change in the energy easily.

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  • $\begingroup$ Thank you! You meant "the potential energy stored", right? $\endgroup$ – evaristegd Sep 24 '18 at 3:28
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    $\begingroup$ @evaristegd: Yes, that's right. $\endgroup$ – user7777777 Sep 24 '18 at 3:29

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