Charge Distribution on Two Neighboring Conductors

[conquerer7]

Say you have two spherical conductors, of radius r, centers a distance d apart. Both have charges of +Q. What are the charge distributions on them?

My physics book more or less handwaves and says it'll be more or less unaffected, but is there a way to solve this exactly?

 

[Naty1]

I haven't solved an problems like that in many,many years...but I can provide a bit of an insight:

are these solid or hollow conductors...?

anyway, Say d is much, much greater than r...there should be a uniform distribution on each sphere...now bring them close together, say d is just a bit bigger than 2 r so they are close...like charges will repel somewhat at the closest point, less so further...so it sure
seems like the charge distribution WILL change...I disagree it would be "unaffected".

 

[Meir Achuz]

This problem can be solved with a sequence of image charges. It gets quite complicated, but for d>>a, the series converges fast.