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

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