How Does the Method of Images Determine Potential Inside a Grounded Sphere?

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The discussion focuses on calculating the electrostatic potential inside a grounded conducting sphere with a charge placed outside it. The method of images is used to show that the potential inside the sphere is equivalent to that produced by the actual charge and an image charge located at a specific distance. The potential is calculated for the region where the distance from the sphere's center is less than the distance to the image charge. The challenge lies in incorporating Legendre polynomials into the solution, as the initial approach using simple potential equations does not align with the homework's focus on Legendre topics. The user seeks guidance on how to integrate these concepts effectively.
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Homework Statement


We have a grounded conducting sphere of radius R and a charge q is placed a distance a from the center of the sphere. Show that the potential in the interior produced by q and the distributed induced charge is the same as that produced by q and its image charge q'.The image charge is at a distance a'=R^2/a fro the center collinear with q and the origin. Calculate the electrostatic potential for a<R<a'. Show that potential vanishes for r=R if we take q'=-qR/a


Homework Equations


In the course we are investigating legendre equations, legendre polynomials etc.


The Attempt at a Solution


I can find potential by simply using V=Sum (1/4pieps)q/r however there is no legendre polynomial or legendre series in it. I tried Laplace's Equation however I got confused. Because in Laplace's eqn I can only deal with r. Where have I gone wrong? How should i think?
 
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I know that solution, however there is nothing related to legendre in it. The question was given to me as a homework for Legendre equation/polynomials topic. I should o something related to that I suppose.
 
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