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You can use the method of images for this, giving you an image charge q' located at a distance x' from the center of the sphere. The answer I get is:

q' = -qa/x

x' = a^2/x

(this doesn't really matter for my question)

Anyway, solving for the total charge induced on the sphere (by evaluating E at r=a, giving sigma, then integrating over the area of the sphere), it took pages and pages of grinding thru the messy integral -which eventually simplifed, and gave a total charge induced on the sphere = q.

When I saw this result I figured that if it simplified to this, it should have been obvious without going thru the integral. If I were to guess, I would have guessed that the total charge on the sphere should hve been q', not q. But even this isn't clear to me.

But anyway, was there a simpler way to show what the TOTAL charge on grounded conductor would be when a charge q sits some distance x away from it? I was thinking of Gauss' law, but that doesn't give me the answer I'm looking for.

Thanks