A dipole with an electric moment p is located at a distance r from a along thread charge uniformly with a linear density `lambda`. Find the force F acting on the dipole if the vector p is oriented
(a) along the thread
(b) along the radius vector r
(c ) at the right angles to the thread and the radius vector r.

Explanation

Here, `E_(r) = (lambda)/(2pi epsilon_(0) r), E_(0) = E_(varphi) = 0` and `vec(F) = p (del vec(E))/(del l)`
(a) `vec(p)` along the thread.
`vec(E)` does not change as the point of observation is moved along the thread.
(b) `vec(p)` along `vec(r)`,
`vec(F) = F_(r) vec(e_(r)) = (lambda p)/(2pi epsilon_(0) r^(2)) vec(e_(r)) = (lambda vec(p))/(2pi epsilon_(0) r^(2))` (on using `(del)/(del r) vec(e_(r)) = 0`)
(c) `vec(p)` along `vec(e_(theta))`
`vec(F) = p (del)/(r del theta) (lambda)/(2pi epsilon_(0) r) e_(r)`
`= (p lambda)/(2pi epsilon_(0) r^(2)) (del vec(e_(r)))/(del theta) = (p lambda)/(2po epsilon_(0)r^(2)) vec(e_(theta)) = (vec(p) lambda)/(2pi epsilon_(0) r^(2))`