A point electric dipole with a moment `p` is placed in the external uniform electric field whose strength equals `E_(0)`. With `p uarr uarr E_(0)`. In this case one of the equipotential surfaces enclosing the dipole from a sphere. Find the radius of this sphere.

Explanation

Let us assume that the dipole is at the centre of the one equipotential surface which is spherical (Fig). On an equipotential surface the net electric field strength along the tangent of it becomes zero. Thus
`-E_(0) theta + E_(theta) = 0` or `-E_(0) sin theta + (p sin theta)/(4pi epsilon_(0) r^(3)) = 0`
Hence `r = ((p)/(4pi epsilon_(0) E_(0)))^(1//3)`
Alternate : Potential at the point, near the dipole is given by,
`varphi = (vec(p). vec(r))/(4pi epsilon_(0) r^(3)) - vec(E_(0)) . vec(r+)` constant,
`= ((p)/(4pi epsilon_(0) r^(3))- E_(0)) cos theta +` Const
For `varphi` to be constant,
`(p)/(4pi epsilon_(0) r^(3)) - E_(0) = 0` or `(p)/(4pi epsilon_(0) r^(3)) = E_(0)`
Thus `r = ((p)/(4pi epsilon_(0) E_(0)))^(1//3)`