Final answer:
The potential of a hollow metal sphere changes as drops of mercury each charged to a potential V fall into it. The potential is initially V when the first drop falls in, and it increases with each drop without exceeding the potential of a single drop. The correct answer is B, V' = V for N=1.
Step-by-step explanation:
The question pertains to the concept of electric potential and charge distribution in electrostatics, a topic covered under Physics. It involves understanding how the potential of a hollow metal sphere changes after N drops of mercury, each with a certain charge and potential, fall into the sphere through a small opening. Since the drops distribute their charge uniformly along the inner surface of the sphere, we can treat the total charge as if it were concentrated at the center of the sphere for the purposes of calculating the potential outside and on the surface of the sphere.
Initially, the sphere is uncharged, so its potential is zero. When the first drop falls in, the entire charge of the drop is distributed over the surface of the sphere, and the potential of the sphere becomes equal to the potential of a single drop, so V' = V for N=1.
As more drops fall in, they add their charge to that already on the sphere, increasing the total charge and thus the potential. However, the sphere's potential doesn't keep increasing indefinitely.
As per Gauss's Law, the increase in potential depends on the total charge divided by the radius of the sphere, which means the potential of the sphere after N drops depends on the formula V'= (Nq)/ (4περR) where q is the charge on one drop, ερ is the permittivity of free space, and R is the radius of the hollow sphere. Therefore, as N increases, the potential V' of the sphere increases in such a way that it can never exceed the initial potential V of a single drop unless the sphere's radius is comparable to the size of a drop, in which case the system would no longer behave as described in the question.
In conclusion, looking at the options provided, and based on the explanation above, the correct answer is option B, V' = V for N=1.