Charge leakage from two suspended charged spheres

Question

Recently I was going through "Problems in General physics" by I E Irodov. In Electromagnetics chapter, there is a question how much is the charge leakage from two spheres suspended by a silk thread (3.3).

Two small equally charged spheres, each of mass $m$, are suspended from the same point by silk threads of length $l$. The distance between the spheres $x \ll l$. Find the rate $\frac{dq}{dt}$ with which the charge leaks off each sphere if their approach velocity varies as $v = \frac{a}{\sqrt{x}}$, where $a$ is a constant.

$$ \frac{1}{4 \pi \epsilon_0} \frac{q^2}{x^2} = \frac{mgx}{2l} . $$

Take derivative then, $$ {2q} \frac{dq}{dt} = \frac{4\pi \epsilon_0 mg}{2l} 3 x^2 \frac{dx}{dt} . $$

One may simplify further by introducing expression for q from the electrostatic force equation and may do more arithmetic... For my point here, this much is enough, it is clear that dq/dt depends on dx/dt (it is vivid in the question too).

Here I did not understand the concept fully. I have two questions:

1. What charge leakage means? Where the charge is leaking to?

2. The expression for the charge leakage dq/dt depends on the relative velocity between the two charged spheres. If the spheres are held stationary or if the relative velocity is zero, according to the equation no charge leakage happens, right? Can any one explain physically why no charge leakage happens when relative velocity is zero?

Answer 1

To your questions:

(1) Charge leakage $dq/dt$ is the charge loss current of the spheres due to the conductivity of the air.

(2) If the spheres are losing charge, the repulsive Coulomb force decreases and thus also the distance x decreases. If the spheres maintain a constant distance x, while they are kept apart by the repulsive Coulomb force, then there is no charge loss. That "no charge leakage happens when the relative velocity is zero" is not the causal sequence. The correct cause-effect relation is converse: the relative velocity is zero because no charge leakage happens.