Question

A negative charge, -q, has a mass, m, and an initial velocity, v, but is infinitely far away from a fixed large positive charge of +Q and radius R such that if the negative charge continued at constant velocity it would miss the center of the fixec charge by a perpendicular amount b. But because of the Coulomb attraction between the two charges the incoming negative charge is deviated from its straight line course and attracted to the fixed charge and approaches it. Find the closest distance the negative charge gets to the positive one.

Try to use work-energy theorem with U=(kq)/r and KE=(mv^2)/2

Answer 1

First apply angular momentum conservation about the centre of the fixed positive charge then apply energy conservation theorem (or, work-energy theorem ), considering kinetic energy of negative charge and electrostatic Forces .

Here we go,

Angular momentum conservation : mvb = mv'b' (v' -final velocity at minimum dist., b' - minimum distance) ->(eqn 1.)

Engery cons. : 1/2(mv*v) -1/2(mv'*v') =1/2(kq*Q/b') -> (eqn. 2.)

So we have two variables and two equation ,it's now easy to solve .

Also since positive charge is fixed ,it's angular momentum is anyways zero about its centre and it's kinetic energy is also zero all time .

For eqn 1., Here you cannot apply linear momentum conservation as positive charge is fixed and we dont know the unknown forces holding the charge in its place and net torque of all those unknown forces is zero about it's centre , and the torque of electrostatic forces between the charges is also zero as it acts along the central line . Therefore, we applied angular momentum conservation .

For eqn 2., And we know here, work done by electrostatic forces between the charges is equal to the change in kinetic energy.