Small oscillations and a time dependent electric field

[Shivy G]

Homework Statement

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Here's the problem from the homework. I've called the initial positions in order as 0, l, and 2l.

Homework Equations

The most important equation here would have to be
|V - w²*M| = 0,
where V is the matrix detailing the potential of the system and M as the "masses" of the system. This eigenvalue equation would normally be solved after getting a V and M to find the eigenfrequencies, and the normal modes from there. However, this problem is asking for the individual positions, so I'm guessing normal modes isn't the way to go unless we are to parse out the equations for the individual positions from the normal modes.

Another equation is E = -grad(V), since we could get the electric potential from our electric field and from that, the electrical potential energy depending on the position of the particles.

However, I don't feel like the eigenvalue equation will help much since the potential is time dependent. Maybe I'm wrong about this.

The Attempt at a Solution

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My first attempt was to just write out the potential based on the springs and the electric field. However, I realized the issue with this method is that the time dependence of the electric field will complicate things, and I'll get the normal modes instead of the individual positions.

My next attempt was to write out Newton's equations for each of the individual particles, but these equations turned out to be too complicated to solve in a straightforward manner.

I'm wondering what the best way to proceed on this question would be given an understanding of small oscillations (this is a problem from a PSET on small oscillations). Any help would be greatly appreciated!

 

[TSny]

For this particular problem, I would recommend going with the equations of motion (Newton's equations).

What can you say about the net external force acting on the system? How can this observation help with solving the equations of motion?

 

[Shivy G]

The net external force on the system should be zero since the individual forces of the charges due to the electric field cancel (the net charge is 0). This implies the center of mass doesn't move, which would give us another equation for the positions that we could make use of.

 

[TSny]

The net external force on the system should be zero since the individual forces of the charges due to the electric field cancel (the net charge is 0). This implies the center of mass doesn't move, which would give us another equation for the positions that we could make use of.

Yes, good.

 

[Shivy G]

Awesome! I'll give it a shot.