Graduate How Do Nested Conducting Loops Interact Electromagnetically?

[bob012345]

Hello,

As a heuristic tool for a complex problem, I'm trying to understand a series of nested conducting loops each with a initial identical current on it. These perfectly conducting loops are in the same plane each with that current in either direction. Each nested loop is perfectly conducting but also perfectly elastic, it can grow or shrink and only has electromagnetic forces.

Starting with one loop, we add another loop and try and figure out how the first loop changes and what the final configuration is that minimizes overall energy. They can merge into one loop or remain separate loops.

What my conceptual issues are is when we get beyond a couple loops and one has induced currents and forces between the loops. I just want to get a handle on a systematic process of understanding what one might call paramagnetic or diamagnetic forces, depending on whether they attract or repel for each new loop and how each new loop effects the previous loops.

The magnitude of starting currents in each loop is fixed but can be in either direction but once set for a loop doesn't change.

I just need hints at a systematic approach. Thanks.

 

[mfb]

Simulate it in small time steps? For a given configuration at a given time, you can determine the forces, and translate them to accelerations (or velocities if you like a lot of friction).

 

[bob012345]

Simulate it in small time steps? For a given configuration at a given time, you can determine the forces, and translate them to accelerations (or velocities if you like a lot of friction).

Thanks but I'm thinking of setting up an equation to determine the equilibrium situation. It's things like how the currents are reflected and induced (if that happens) that stump me. It's how to determine the forces that act. For example, I can see that the second loop will have a magnetic field that effects the first loop. Depending on the currents, it will either repel or attract. I assume circular symmetry. There must be an equal and opposite force on the other loop.

 

[mfb]

Technically you could have some induction if things are moving quickly, but not in the equilibrium state. For the equilibrium states you get static conditions, you can set up equations for the equilibrium and solve them. I would be surprised if there is a stable equilibrium at all. The outer loop should expand forever, or collapse until it hits an inner loop which then expands further to become an expanding outer loop.

 

[bob012345]

Technically you could have some induction if things are moving quickly, but not in the equilibrium state. For the equilibrium states you get static conditions, you can set up equations for the equilibrium and solve them. I would be surprised if there is a stable equilibrium at all. The outer loop should expand forever, or collapse until it hits an inner loop which then expands further to become an expanding outer loop.

Thanks. I agree with you although for simplicity I left out part of the problem that would stabilize the loops since I'm not looking for a complete solution yet. I agree about no induction at equilibrium. We could assume they are rigid for now and there will be induction as things work toward equilibrium each time we add a current loop but at equilibrium there could be stable currents that are reflections of the original currents induced during the period of change. We are adding energy each time we add a new current loop.