Acidified water from certain reservoir kept at a potential $$V$$ falls in the form of small droplets each of radius $$r$$ through a hole into a hollow conducting sphere of radius $$a$$. The sphere is insulated and is initially at zero potential. If the drops continue to fall until the sphere is half full, the potential acquired by the sphere is

A

$$\frac{a^2 V}{2r^2}$$

B

$$\sqrt{\frac{a}{r}}\,\frac{V}{2}$$

C

$$\frac{a^3 V}{2r^3}$$

D

$$\frac{aV}{r}$$