Question

Can spheres leaking charge be assumed to be in equilibrium?

Answer 1

Two small equally charged spheres, each of mass mm, are suspended from the same point by silk threads of length ll. The distance between the spheres x≪lx≪l. Find the rate dqdtdqdtwith which the charge leaks off each sphere if their approach velocity varies as v=ax√v=ax, where aa is a constant.

This is embarrassingly simple; we make an approximation for x≪lx≪l and get

14πϵ0q2x2−mgx2l=mx¨.14πϵ0q2x2−mgx2l=mx¨.

We can get x¨x¨ from our relation for vv, so we can solve for qq and then find dqdtdqdt.

However, in general, dqdtdqdt will depend on xx and hence on tt. The answer in the back of the book and other solutions around the web have dqdtdqdt a constant.

You can get this by assuming that at each moment the spheres are in equilibrium, so that you have x¨=0x¨=0 in the equation of motion above.

Does the problem tacitly imply we should assume equilibrium and hence dqdtdqdt is constant, or am I missing something entirely? I.e. why is the assumption of equilibrium justified? I understand reasoning like "the process happens very gradually, so the acceleration is small compared to other quantities in the problem," but I don't understand how that is justified by the problem itself, where we are simply given that the spheres are small (so we can represent them as points) and x≪lx≪l (which we have used to approximate the gravity term in the equation of motion).

Answer 2

Hey !

______________________________

The leaking charge is assumed as the zero in the equillbrium.

Conductivity theory is taken as the perfect solution

______________________________

Thanks !