A droplet of liquid metal has constant mass density and constant surface tension σ, which causes it to form into a sphere of radius R. (Throughout this problem, you may neglect gravity.) A thin wire is inserted into the droplet and connected to an electric current source which slowly charges the droplet. There is a critical value of the charge, Q₀, that causes the droplet to split in half. Each half takes half the total charge, Q₀/2, and half the mass of the original droplet. The "ejected half" is repelled far away from the other half, which remains in contact with the wire.
a. For simplicity, assume the droplet splits as soon as the final state (after the droplet has split in half and the two halves are well separated) has a lower total energy than that of the initial, single droplet. What is the value of Q₀? Give your answer in terms of a dimensionless constant A multiplied by a product of powers of σ, R, and the vacuum permittivity ε₀, and give the numeric value of A to three significant figures.
b. As more charge is added to the droplet by the current source, it continues to split in half repeatedly. What is the charge qₙ on the nth ejected droplet? Give your answer in terms of Q₀.
c. In the limit where all of the initial mass of the droplet has been ejected, what is the total work done by the current source? Give your answer in terms of a dimensionless constant B multiplied by a product of powers of σ, R, and ε₀, and give the numeric value of B to three significant figures.