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I was able to solve option A conceptually- at centre of sphere, potential due to induced charge on sphere is 0 as left and right hemispheres' induced charge is same and opposite. So only potential due to point charge needs to be calcuated.

But I am not getting how to solve for option D. Do I calcuate the induced charge on the sphere (which is not same throughout), or can I solve using gauss theorem, or any other method which I could not think of?

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1) Since, the potential at center of neutral conducting sphere is $\cfrac{q}{4\pi\epsilon o(d +r)}$ as you mentioned.

2) Now it is a well known concept that potential throughout the conductor remains constant so each and every point (including B) throughout the conductor should have same potential value which is mentioned in point 1)

3) now point at B-

$V_{q}$ + $V_{sphere}$ $=$ $\cfrac{q}{4\pi\epsilon o(d +r)}$ ----$\mathrm{I}$

since $V_q$ at point B is $\cfrac{q}{4\pi\epsilon o d}$, putting it in $\mathrm{I}$

$V_{sphere}$ $=$ $\cfrac{-qr}{4\pi\epsilon o(d + r)d}$

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