Equilibrium Shape of a Charged Elastic Ring

[xanadu77]

Suppose we have a necklace made of a conducting material. We join the two ends and leave it on a frictionless non-conducting table. Then we charge it negatively. What is the equilibrium shape of the necklace? The answer to this is probably a circle. I am actually looking for the differential equation governing the dynamics of this necklace.

Here's a (probably) simpler question posed more mathematically: We charge a non-self-intersecting closed curve on the plane negatively. The curve can be any closed curve. Let the modulus of elasticity, length and total charge be given. I am looking for the differential equation for this problem. Other results are also welcome, such as the tension in the curve at equilibrium.

The differential equation is probably too complex. Book, article etc. suggestions are also welcome.

 

[Eynstone]

Suppose we have a necklace made of a conducting material. We join the two ends and leave it on a frictionless non-conducting table. Then we charge it negatively. What is the equilibrium shape of the necklace? The answer to this is probably a circle. I am actually looking for the differential equation governing the dynamics of this necklace.

Here's a (probably) simpler question posed more mathematically: We charge a non-self-intersecting closed curve on the plane negatively. The curve can be any closed curve. Let the modulus of elasticity, length and total charge be given. I am looking for the differential equation for this problem. Other results are also welcome, such as the tension in the curve at equilibrium.

The differential equation is probably too complex. Book, article etc. suggestions are also welcome.

When the ring has a radius r, let U(r) be the elastic energy & V(r) be the electric potential energy of the ring ( you can calculate this as the charge & the modulus of elasticity is given). Solve d/dr ( U(r) + V(r)) =0 for the equilibrium r.

 

[AJ Bentley]

Hmmm...
Tricky.

I read that as something like an infinitely thin, simple loop (no self-crossings) constrained to two dimensions (why not make it three?) and obeying both Hookes law for some arbitrary constant of elasticity and Coulombs law for an arbitrary distribution of free-flowing charge over the length of the loop?

It probably isn't solvable by analytic means. The charge static distribution problem alone on an arbitrary shaped conducting loop would be a nightmare problem. It would require a boundary-value solution to Laplace's Equation with an unknown initial boundary condition. Then you want the boundary to move under other constraints?

You could start from the 'it's probably a circle' position and see what happens from there but I suspect it would become chaotic.

Do you have a specific problem in mind?

 

[xanadu77]

Thanks for the replies. Actually no, I asked just out of curiosity.

Perhaps the best way is actually putting a charged necklace on a table and observing the behavior :)

But I would guess someone had at least written the equations in some book or article. It is an easy problem to think of, but difficult to solve.

 

[AJ Bentley]

Doubtful.
Feynman V2 7-2

 

[xanadu77]

This problem actually came up during a discussion about "is there a perfect circle in the nature, or can it be constructed?" I came up with this answer, thinking that an elastic charged ring would assume the shape of a circle in equilibrium. But now that I realize the complexity of the problem, this may not be the case. And probably much simpler constructions of a perfect circle exist :)