PX157 - B8d - potential of an infinite line of charge

\vec{E} = \frac{λ}{2 π ϵ_{0} R} \hat{R}

\begin{aligned} V_{a} & = \int_{R_{a}}^{R_{b}} \frac{λ}{2 π ϵ_{0} R} \hat{R} \cdot \hat{R} d R \\ = \frac{λ}{2 π ϵ_{0}} \int_{R_{a}}^{R_{b}} \frac{1}{R} d R \\ = \frac{λ}{2 π ϵ_{0}} \ln [\frac{R_{b}}{R_{a}}] \end{aligned}

V_{a} = \frac{λ}{2 π ϵ_{0}} \ln [\frac{R_{0}}{R_{a}}]

∴ V (R) = \frac{λ}{2 π ϵ_{0}} \ln [\frac{R_{0}}{R}]