Derive an expression for electric potential energy of a systemm of charges in an electric field.

Explanation

Consider an electric dipole in an uniform electric field `vec(E)` with its dipole moment `vec(p)` making an angle `theta` with the field as shown in fig.
The torque acting on the dipole is
`tau = pE sin theta` ………(1)
If the dipole is rotated through a small angle `d theta` against the torque acting on it, the small amount og work done during this process is
`d W = tau d theta` .......(2)
From (1) `dW = p E sin theta d theta` .........(3)
`therefore` The total amount of work done in rotating the dipole from its orientation `theta_(1)` to `theta_(2)` is
`W = underset(pi//2)overset(theta)int pE sin theta`
`= pE [-cos theta]_(pi//2)^(theta)`
`= pE[-cos theta - cos""(pi)/(2)] " " | therefore cos""(pi)/(2) = 0`
`W = -pE cos theta` ..........(3)
This work done is stored as the electrostatic potential energy of the dipole
`therefore U = -pE cos theta`
`U = -vec(p).vec(E)`