Solving Difficult E&M Problem: Two Strings Attached to Balls

[mfig]

Homework Statement

Two flexible but not extensible strings with linear charge density s and length L are attached to two insulating balls a distance D apart, where D<L. What do the strings look like when the system is at equilibrium?

Homework Equations

The Attempt at a Solution

I think that the strings will look like two circular arcs, but I am not sure about that. I was thinking that they would lie on the equipotential lines as if the distance D was itself a line charge, but I don't know why. Is it true that the tension in the string would have to balance out the es force at each string element? I was wondering if writing the lagrangian for the problem and integrating would help, but I want to know if anyone has any insight as to how to tackle this using some nifty "trick" based on symmetry or something. Thanks.

 

[pam]

The curves won't be circular arcs.
Did someone assign this problem or did you make it up?
Even with one string, it looks very difficult.

 

[mfig]

Assigned...

Yep, it was assigned. Why are you sure the strings won't be circular arcs? It seems intuitive to me that they are, but I don't know how to go about proving it. Can you suggest a simplifying approach?

 

[pam]

I thought that each string was tied to each ball.
Is the configuration different than that?
Sorry, I don't see any simplification for any case.

 

[jeff_m]

spamanon,

Several things to consider: the problem states that the charge density/length is constant. That makes the problem much easier. The other thing is that there are 2 strings, so there is symmetry along the axis joining the 2 insulated spheres. Since nothing about the insulating balls is given in terms of Er or anything, the two balls can be assumed to be out of the problem other than the fact of their separation distance. The problem then reduces to an energy minimization problem that can be solved via variational principles. Look at Goldstein's "Classical Mechanics" and the brachristochrone problem. The only difference is that the force acting on the strings will not be constant per unit length as is the case for gravity. You can take advantage of symmetry to simplify the expression for computing the energy by noting that along the axis of symmetry the potential is zero. You will need to solve for the equation of the string that yields the minimum energy of the system with the constraint that the total length of the string is fixed and greater than the distance between the two insulating spheres. Remember also that the minimum energy configuration requires contributions from between the 2 strings as well as the self contribution of each string.


Good luck.

Jeff

 

[mfig]

Thanks jeff_m

This is kinda what I was thinking. Now I am wondering if I should formulate the question this way: a charged particle traveling on the path (Ball 1 to Ball 2) that has a fixed length (L) but that minimizes the work done on it. Are these two problems the same? Or is it that I find the shape of the strings that minimizes the potential energy of the whole system.
I understand that I will have to use the Euler equations, but I am stuck on how to formulate the problem in terms of what I should actually minimize. Thanks.

 

[jeff_m]

spamanon,

You want to minimize the entire energy in the system as defined by the double integration over each infinitesmal segment of one string (see Jackson: "Classical Electrodynamics" for a good explanation of how to compute the energy.) Once you have this set up then you can apply the Euler equatons subject to the constraints of string length and its being fixed at both ends.

Jeff