PX157 - B8f - potential of an ideal electric dipole

origin is on the midpoint of the two charges, $+ q$ and $- q$

\begin{aligned} \vec{E} & \approx \frac{1}{4 π ϵ_{0} r^{3}} [3 (\vec{p} \cdot \hat{r}) \hat{r} - \vec{p}] \\ \vec{E} & \approx \frac{p}{4 π ϵ_{0} r^{3}} (2 \cos θ \hat{r} + \sin θ \hat{θ}) \end{aligned}

$\vec{p} = q \vec{l} = q l \hat{z}$
for the path: $\vec{r} = r \hat{r} ⟹ d \vec{l} = d \vec{r} = d r \hat{r}$
apply $V_{a} - V_{b} = \int_{a}^{b} \vec{E} \cdot d \vec{l}$ :

\begin{aligned} V_{a} & = \int_{r_{p}}^{r_{b}} \frac{p}{4 π ϵ_{0} r^{3}} (2 \cos θ \hat{r} + \sin θ \hat{θ}) \hat{r} d r \\ = \frac{p \cos θ}{2 π ϵ_{0}} \int_{r_{p}}^{r_{b}} \frac{1}{r^{3}} d r \\ = \frac{p \cos θ}{2 π ϵ_{0}} {(- \frac{1}{r^{2}})}_{r_{p}}^{r_{b}} \\ = \frac{p \cos θ}{2 π ϵ_{0}} (\frac{1}{r_{p}^{2}} - \frac{1}{r_{b}^{2}}) \end{aligned}

define $V (r_{b} \to \infty) = 0 :$

V_{a} = \frac{p \cos θ}{2 π ϵ_{0} r_{p}^{2}}

true for any value of $r_{p} :$

∴ V (r) = \frac{p \cos θ}{2 π ϵ_{0} r^{2}} = V_{a} = \frac{\vec{p} \cdot \hat{r}}{2 π ϵ_{0} r^{2}}

eg: electric field that is discontinuous between $R = R_{1} \to R_{2}$
- can only choose one point where $V = 0$
$V_{a} - V_{b} = \int_{R_{a}}^{R_{b}} \vec{E} \cdot d \vec{l} = \int_{R_{a}}^{R_{2}} \vec{E} \cdot d \vec{l} + \int_{R_{2}}^{R_{1}} \vec{E} \cdot d \vec{l} + \int_{R_{1}}^{R_{b}} \vec{E} \cdot d \vec{l}$
note: potential is always continuous