$(a)\;\large\frac{k\;q}{2}\qquad(b)\;-\large\frac{2\;k\;q}{4}\qquad(c)\;-\large\frac{k\;q}{2}\qquad(d)\;zero$

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Answer : (d) zero

Explanation :

Let $\;P (x , y ,z)\;$ be any point on the sphere . From the property of the sphere

$x^2+y^2+z^2=4^2=16$

Further ,

$PA=\sqrt{(x-2)^2+y^2+z^2}$

$=\sqrt{x^2+y^2+z^2+4-4x}$

$=\sqrt{16+4-4x}$

$PA=\sqrt{20-4x}$

and $\;PB= \sqrt{(x-8)^2+y^2+z^2}$

$=\sqrt{x^2+y^2+z^2+64-16x}$

$=\sqrt{80-16x}$

$=2 \sqrt{20-4x}$

Let $\;V_{P}\;$ be potential at P

$V_{P}=\large\frac{1}{4 \pi \in_{0}}\;[\large\frac{q}{PA}-\large\frac{2\;q}{PB}]$

$=\large\frac{1}{4 \pi \in_{0}}\;[\large\frac{q}{\sqrt{20-4x}}-\large\frac{2\;q}{2 \sqrt{20-4x}}]$

$=0\;.$

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