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I was wondering how it would be possible to solve for the electric potential inside of a metal sphere that has a total charge Q without using Gauss's Law? I completely understand the problem with Gauss's law but am not quite sure how to solve it without Gauss's Law? Can someone please help explain this I really want to understand how to do it without Gauss's Law?

Thank you and sorry for any missing tags, feel free to edit.

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    $\begingroup$ This is basically Newton's shell theorem. Although Newton derived his theorem for gravity it applies to any inverse square law force. $\endgroup$
    – John Rennie
    Dec 12, 2021 at 20:16

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Use the integral equation for potential, Where r' is the position vector on a sphere, This is easily done using spherical coordinates with the spherical Dv, element. The distance from a point on the sphere to a point along my axis is just found easily using the law of cosines, This answer is finding the potential on the z axis, however using symetry it is generalised to any point.

This answer is for a hollow sphere, to generalise this for a solid sphere, Change R (radius of sphere) to r the radial position, and integrate over dr aswell as da https://prnt.sc/22rv6pl

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  • $\begingroup$ Thank you, understood! $\endgroup$
    – harry
    Dec 13, 2021 at 3:08
  • $\begingroup$ I would advise you to actually go through this integral and compute it and be carfeful when simplifying the expression under the square root,(Sqrt(x))^2 = abs(x) not x. You need to split the integral into 2 different integrals for when going inside the sphere! $\endgroup$
    – jensen paull
    Dec 13, 2021 at 10:01

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