2.2C: Long Charged Rod

The field at a distance \(r\) from a long charged rod carrying a charge \(λ\) coulombs per meter is \(\frac{\lambda}{2\pi\epsilon_0 r}\). Therefore the potential difference between two points at distances \(a \text{ and }b\) from the rod \((a < b)\) is

\[\begin{align}V_b-V_a&=-\frac{\lambda}{2\pi\epsilon_0}\int_a^b \frac{dr}{r}. \\ \therefore V_a-V_b&=\frac{\lambda}{2\pi\epsilon_0}\ln (b/a).\tag{2.2.6} \\ \end{align}\]