Answer to Question #292180 in Electricity and Magnetism for Shah

Question

The coordinate-free form of electric field at an arbitrary point P (a distance r away). of an electric dipole located at the origin is E(r)= 1 4 pi epsilon 0 [ 3(p.r) r^ 3 r- p r^ 3 ] Convert this into an expression in spherical coordinales.

Answer 1

$E(r)=\frac1{4\pi\varepsilon_0}\frac 1{r^3}(3(\vec p \cdot\vec r)\vec r-\vec p),$

$E(r,\theta)=\frac p{4\pi\varepsilon_0r^3}(2\cos\theta \vec r+\sin\theta\vec\theta).$