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I think I remember the formula for its electric potential energy is ##U = \displaystyle {Q^2 \over 2R}##.

Now I want to derive it. I used ##\displaystyle U =\displaystyle\dfrac{1}{8\pi}\int_{\text{Region}} \vec E\cdot \vec E\ \ dv##.

I know that electric field by an sperical shell is ##Q\over R^2##.

So I got,

##\displaystyle U =\displaystyle\dfrac{1}{8\pi}\int_{\text{Region}} {Q^2\over R^4}\ \ dv = \displaystyle\dfrac{Q^2 }{8\pi}\int_{\text{Region}} {1 \over R^4 }\ \ dv##

Then I used ##v = {4\pi \over 3}R^3## to get ##\displaystyle \left({3v \over 4 \pi}\right)^{1/3} = R##

Back in integral, ##U = \displaystyle\dfrac{Q^2 }{8\pi}\int_{\text{Region}} \left({4 \pi \over {3v} }\right)^{4/3}\ \ dv##

For limits ##v \to \infty## I got the desired answer.

Is this correct ? I have some confusions because the proofs I saw on internet involve triple integral and spherical coordinates.

I think I am doing something wrong as why would anybody use spherical coordinates for such a simple proof ?