Question

A point dipole with an electric moment $p$ oriented in the positive direction of the $z$ axis is located at the origin of coordinates. Find the projections $E_z$ and $E_{\perp }$ of the electric field strength vector (on the plane perpendicular to the $z$ axis at the point $S$ (see Fig.). At which points is $E$ perpendicular to $p$?

Answer 1

From the results,

$E_r=\frac{2p\cos \theta }{4\pi {ϵ}_0{r}^3}$ and $E_{\theta }=\frac{p\sin \theta }{4\pi {ϵ}_0{r}^3}$

From the given figure, it is clear that,

$E_z=E_r\cos \theta -E_{\theta }\sin \theta =\frac{p}{4\pi {ϵ}_0{r}^3}(3{\cos }^2\theta -1)$

and $E_{\perp }=E_r\sin \theta +E_0\cos \theta =\frac{3p\sin \theta \cos \theta }{4\pi {ϵ}_0{r}^3}$

When $\overrightarrow{E}\perp \overrightarrow{p},|\overrightarrow{E}|=E_{\perp }$ and $E_z=0$

So $3{\cos }^2\theta =1$ and $\cos \theta =\frac{1}{\sqrt{3}}$

Thus $\overrightarrow{E}\perp \overrightarrow{p}$ at the points located on the lateral surface of the cone, having its axis, coinciding with the direction of z-axis and semi vertex angler $\theta ={\cos }^{-1}\left(1/\sqrt{3}\right)$.