(a) Derive an expression for the elecrtric field E due to a dipole of length '2a at a point
distant r from the centre of the dipole on the axial line.
(b) Draw a graph of E versus r for r `gt gt a`.
(c) If the diploe were kept in a uniform external electric fields `E_(0)` Diagrammatically represent
the position of the dipole in stable and unstable equilibrium and write the expressions
for the torque acting on the dipole in both the cases.

Explanation

(a) Let there be an electric dipole consisting of charges -q and + q , separated by distance 2n
and placed in vacuum . Let P be a point on the axial line at a distance r from centre O of
the dipole on the dipole on the side of the charge `+ q . overset(to)(E)` due to charge - q at P is
` overset(to)(E)_(-q) = (-q)/(4piin_(0) (r + a)^(2)) hatP` (towards left) (along AB)
Where, `hatP` unit vector along dipole axis from q to q
` overset(to)(E)_(-q) ` due to charge + q at P is
` overset(to)(E)_(-q) = (-q)/(4piin_(0) (r + a)^(2)) hatP` (along BP)
` overset(to)(E)_("axial")= overset(to)(E)_(+q) + overset(to)(E)_(-q)`
` overset(to)(E)_("axial")= (q)/(4piin_(0))= [(1)/((r- a)^(2)) - (1)/((r+ a)^(2)) ] hatP`
` overset(to)(E)_("axial")= (q)/(4piin_(0))= (4ar)/((r^(2) - a^(2)) )hatP = (1)/(4 piin _(0))xx (2Pr)/((r^(2) - a^(2))) hatp ` ( Directed towards `bar(BP)` product.)
` P = 2 a xx q ` = Dipole moment
For ` r gt gt a , a^(2)` is neglected
`rArr overset(to)(E)_("axial")= (1)/(4 pi in _(0)) (2P)/(r^(3))` (towards right)
(b) ` E = (1) /(4 pi in _(0))xx (2p)/(r^(3))`

(c) Torque `tau = PE sin theta`
In stable equillbrium ` theta = 0`
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