Exercise 3.3.4: Voltage of a two-wire line

For one infinite line charge $E={\rho }_L/(2\pi {\epsilon }_0\rho )a_{\rho }$ (Problem 3.2.1), so between two radii

(The reference cannot be at infinity: for an infinite line the potential grows without bound there.) Superposing the potentials of $+{\rho }_L$ and $-{\rho }_L$,

and $V=0$ on the mid-plane, where ${\rho }_1={\rho }_2$, gives $C=0$:

On the surface of the positive wire ${\rho }_1=a$ and, since $D\gg a$, ${\rho }_2\simeq D$. By symmetry the negative wire is at the opposite potential:

With $\pi {\epsilon }_0=2.778\times 10^{-11}F/m$ and $\ln 200=5.298$,

On the mid-plane ${\rho }_1={\rho }_2$ and $V=0$, near the positive wire (${\rho }_1\to 0$) $V$ is positive and large, and the unit of ${\rho }_L/{\epsilon }_0$ is V. Numerically $10^{-8}/(2.778\times 10^{-11})=360.0$, times $5.298$, gives $1907\mathrm{V}$.