A point dipole with an electric moment `p` oriented in the positive direction of the z axis si located at the origin of coordinates. Find the projections `E_(z)` and `E_(_|_)` of the electric field strength vector (on the plane perpendicular to the z axis at the poiny `S`. At which points is `E` perpendicular to `p` ?

Explanation

From the results, obtained in the previous problem,
`E_(r) = (2P cos theta)/(4pi epsilon_(0) r^(3))` and `E_(theta) = (p sin theta)/(4pi epsilon_(0) r^(3))`
From the given figure, it si clear that,
`E_(z) = E_(r) cos theta - E_(theta) sin theta = (p)/(4pi epsilon_(0) r^(3)) (3 cos^(2) theta - 1)`
and `E_(_|_) = sin theta+ E_(0) cos theta = (3 p sin theta cos theta)/(4pi epsilon_(0) r^(3))`
When `vec(E) _|_ vec(p), |vec(E)| = E_(_|_)` and `E_(x) = 0`
So `3 cos^(2) theta = 1` and `cos theta = (1)/(sqrt(3))`
Thsu `vec(E_(_|_)) vec(p)` at the points located on teh lateral surface of the lateral surface of the come, having its axis, coinciding with the direction ofz-axis and semi vertex angle `theta = cos^(-1) 1//sqrt(3)`