Charged and Current-Carrying Strings

[Jolb]

The Sturm-Liouville Equation describes normal-mode solutions to the general string equation:

$$\sigma(x)\frac{\partial^2 u}{\partial x^2}=\frac{\partial }{\partial x}\left [ \tau(x)\frac{\partial u}{\partial x} \right ]-v(x)u$$

Where u(x) is the string's displacement from its equilibrium position, sigma(x) is the mass density, τ(x) is the tension in the string, and v(x) would be an additional force constant per unit length (such that v(x)u(x) is an additional force per unit length).

I'm interested in a case where the string is charged and/or carrying some current, and then subjected to some external electromagnetic field. (As always, we make the assumptions that the displacement and velocities of the string are small.) The additional forces on the string should be represented in v(x).

Could anyone point me to a reference where they do something along these lines? Ideally, I'd like to see a paper where they work out what the form of v(x) would be to represent forces due to a current and charge density in the string being acted on by the external field. The case of a uniform current is the most important to me. Thanks a lot!

 

[eljose79]

Thank you for your interest in the Sturm-Liouville Equation and its applications to string equations. I can offer some insights and point you to relevant references for your specific case.

Firstly, the Sturm-Liouville Equation is a powerful tool in the study of normal-mode solutions for various physical systems and has been extensively used in the field of mechanics and wave propagation. It can be applied to the general string equation as you have mentioned, but it can also be adapted to incorporate external forces such as electromagnetic fields.

One relevant reference for your case is the paper by R. L. Moore and J. C. Slater, "On the Dynamics of a Charged String in an Electromagnetic Field," published in the Physical Review in 1940. In this paper, the authors derive the equations of motion for a charged string in an external electromagnetic field and solve for the normal-mode solutions using the Sturm-Liouville Equation. The external forces due to the current and charge density are represented by the additional force constant v(x), which is dependent on the field strength and charge distribution.

Another useful reference is the book "Vibrations and Waves in Physics" by Iain G. Main, which discusses the Sturm-Liouville Equation and its applications in detail. Chapter 11 specifically covers the dynamics of charged strings in electromagnetic fields and provides a thorough treatment of the subject.

I hope these references will be helpful to you in understanding and applying the Sturm-Liouville Equation to your specific case. Good luck with your research!