Question

derive an expression for the potential energy of an electric dipole in uniform electric field

Answer 1

want to find out the potential energy of a electric dipole in a uniform electric field by another process which gives the result to U=−P.EU=−P.E.
Suppose A, B are the position of the point charges qq and −q−qand seperated by small distance D and O be the position of the origin. The distance from O to −q−q is RR.

So potential energy, U=−q.v(R)q.v(RD)U=−q.v(R)q.v(RD)

where v means eletric potential.

v(RD)v(RD) is extended by taylor's series as v(RD)=v(R)Dv(RD)=v(R)D .

del(v(R)) [as |D|<<|R| so we can neglect the smallest terms] putting this value of v(R D),

we get U=qP⋅Δ(v(r))=−P⋅EU=qP⋅Δ(v(r))=−P⋅Ewhere capital letter refers to vector.