Derive an expression for the potential energy of an electric dipole placed in a uniform electric field.

Explanation

1. Consider a electric dipole of length 2a having charges `+q and -q`.
2. The electric diple is placed in uniform electric field E and it's axis makes an angles `theta` with E.
3. Force on charges are equal but opposite sign. They constitute torque on the dipole.
Torque `tau=` one of its force `(F)xxbot^(r)` distance (BC).
`F=qE and sin theta =(BC)/(2a)rArr BC=2a sin theta`
`therefoer` Torque `tau = q E xx2a sin theta = PE sin theta [ therefore p =2aq]`
4. Suppose the dipole is rotated through an angle `dtheta`, the workdone dw is given by `dw= tau d theta =PE sin theta d theta`.
5. For rotating the dipole from angle `theta_1` to `theta_2`, workdone `W=int_(theta_1)^(2^theta)PE sin theta d theta = PE (cos theta_(1)-costheta_(2)`
6. This workdone (W) is then stored as potential energy (U) in the dipole. `therefore U=PE(cos theta_1-cos theta_2)`
7. If `theta_1=90^@ and theta_2=0^@, U=-PEcos theta`.
In vector form `U=- vec(P).vec(E)`.