# Charged and Current-Carrying Strings

1. Apr 8, 2012

### 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!