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Irodov Solutions : Problem 3.3

The statement of the problem is :

Two small equally charged spheres, each of mass m, are suspended from the same point by silk threads having length l. The distance between the spheres x<<l. What is the rate dq/dt with which the charge leaks off each sphere if their approach velocity varies as v=a/√x where a is a constant?

Let's assume for a while that the charge on each sphere is constant (i.e. it is not leaking off the spheres) and hence the spheres are hanging in equilibrium. Writing force equations for one of the spheres,

The value of separation x between the spheres is given by the equation number 3 when each sphere carries a charge q. We can observe that the separation x between the two spheres depends on the charge q. In other words, the separation x is a function of the charge q (or the equilibrium position is the function of charge q which is quite obvious). If q is fixed, x will also be constant. But as stated in the problem, the charge is not fixed; it leaks off from the spheres therefore the separation between the spheres decreases or the equilibrium distance keeps on decreasing. The rate of decrement of the separation or rate of approach of the spheres is given in the statement of the problem as dx/dt = a/√x.

The separation x as a function of q is given by

Thus we get the expression for the rate of leakage of charge from each sphere