Sixty-four identical spherical liquid drops, each carrying the same charge, coalesce to form a single large drop. What is the ratio of the total initial electrostatic energy of all the small drops to the final electrostatic energy of the large drop?

Step-by-step solution

Let the charge on each small drop be q and its radius be r . The electrostatic energy of one small drop is U_ small = kq^2 2r . For N = 64 such drops, the total initial energy is: U_i = N U_ small = 64 kq^2 2r When they coalesce, the total charge of the large drop is Q = Nq = 64q . By conservation of volume, the radius R of the large drop is: 4 3 R^3 = 64 4 3 r^3 R = 64^ 1/3 r = 4r The final electrostatic energy of the large drop is: U_f = kQ^2 2R = k(64q)^2 2(4r) = 4096 4 kq^2 2r = 1024 kq^2 2r The ratio of total