PX157 - B8c - potential of a ring of charges

\vec{E} (x_{p}, 0, 0) = \frac{Q x_{p}}{4 π ϵ_{0} (x_{p}^{2} + a^{2})^{\frac{3}{2}}} \hat{x}

\begin{aligned} \int_{a}^{b} \vec{E} \cdot d \vec{l} & = \frac{Q}{4 π ϵ_{0}} \int_{x_{p}}^{x_{b}} \frac{x}{(x^{2} + a^{2})^{\frac{3}{2}}} d x \\ V_{a} - V_{b} & = \frac{Q}{4 π ϵ_{0}} \frac{1}{\sqrt{x_{p}^{2} + a^{2}}} |_{x_{p}}^{x_{b}} \\ = \frac{Q}{4 π ϵ_{0}} [\frac{1}{\sqrt{x_{b}^{2} + a^{2}}} - \frac{1}{\sqrt{x_{p}^{2} + a^{2}}}] \end{aligned}

∴ V_{a} = \frac{Q}{4 π ϵ_{0}} \frac{1}{\sqrt{x_{p}^{2} + a^{2}}}

d V_{p} = \frac{d Q}{4 π ϵ_{0} \sqrt{x_{p}^{2} + a^{2}}}

\int d V_{p} = V_{p} = \frac{Q}{4 π ϵ_{0} \sqrt{x_{p}^{2} + a^{2}}}