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 [itex]\sigma(z)[/itex] so that
[tex]\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[/tex]
I don't think choosing [itex]\sigma(z) = az[/itex] 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?
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 [itex]\sigma(z)[/itex] so that
[tex]\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[/tex]
I don't think choosing [itex]\sigma(z) = az[/itex] 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?