how to prove this question ?

Question

we have two charged particles on an axis(-3q and -q ) however a third charged particle can be placed at a certain point between the particles. how to prove the equilibrium is stable or unstable ?

Answer 1

"we have two charged particles on an axis(-3q and -q ) however a third charged particle can be placed at a certain point between the particles. how to prove the equilibrium is stable or unstable ?"

A third charged particle (does not matter whether it is a positive or negative charged particle) can be placed in between -3q and -q at x along the joining line to get a stable equlibrium.

Let us put it at x distance from -3q. Then F_1 = k(3qq')/x^2 will be equal to F_2 = k(qq')/(x')^2

Solving for F_1 = F_2 we obtain the

=> ratio of x : x' = sqrt(3) : 1

This make sense since Col Law in inverse square law.

Answer 2

It would be an unstable equilibrium. To understand this question we first need to visualise what the current scenario looks like and what must change to make it an equilibrium state.

Given the two charged particles at the start we see that they are both negative and will repel from eachother. To remedy this we must put a positive charge in the middle that is equally attracted by both negative charges.

Lets say we put a positive charge q a distance x away from the charge -q particle. The Electric Force the particles excert on eachother would be:

Kq²/x²

Now for this to be equal with the other charge, the postive particle must be a distance of, x√3, so that the squaring of the distance in the denominator will cancel with the -3q charge and give us a force of

Kq²/x²

We can now evaluate the fact that both the particle q and the particle -q have the same magnitude of charge, but the charge of -q is a greater distance away from -3q than the positive particle is, meaning the -3q particle will begin moving towards the other two particles and as it does, all the forces begin to change and the system becomes unstable.