Surface Charge on Resistive Current Carrying Cylinder

[MisterX]

The goal is to find a reasonable solution to the surface charge density that would cause constant current through a cylinder of constant resistivity.

One approach is to attempt to find a surface charge distribution on a cylinder so that the electric field on the cylinder axis is constant and non-zero.

We might want to consider this for a cylinder of infinite length. If there are any solutions I would expect a family of solutions, since we could always add a uniform charge distribution and leave the E field unchanged. Essentially I think this problem is find $\sigma(z)$ so that

$$\int_{-\infty}^{\infty} \frac{\mathscr{z}}{(\mathscr{z}^2 + R^2)^{3/2}}\sigma(\mathscr{z} - d) d\mathscr{z} = const \neq 0 \;\;\;\;\forall d$$

I don't think choosing $\sigma(z) = az$ results in convergence (finite E field).

Perhaps a finite sized approach (maybe numerical) is warranted (maybe with charge density on the end caps).

Perhaps the requirement should instead be that the E flux is constant through a cross section of the cylinder.

Does anyone have any ideas?

 

[Vailanor]

One approach is to use the solution of Laplace's equation for a cylindrical surface, which states that the electric potential can be written as a sum of two components:\Phi(r,\theta,z) = \sum_m A_mJ_m(kr)+B_mH_m(kr),where J_m and H_m are Bessel functions, A_m and B_m are constants, and k is a constant related to the resistivity.Using this formula, we can calculate the electric field by taking the derivatives of the potential with respect to r and z. The resulting electric field on the cylinder axis will be a function of z, and we can then solve for the surface charge density that will produce a constant electric field on the cylinder axis.