derive torque(vector)=p(vector) cross E(vector)

by using electric dipole moment concept

dear student,
Consider an electric dipole placed in a uniform electric field of strength E in such a way that its dipole moment vector p makes an angle q with the direction of vector E. The charges of dipole are - q and + q at separation 2l the dipole moment of electric dipole, p = q.2l ...(1)
The forces vector F1 and vector F2 form a couple (or torque) which tends to rotate and align the dipole along the direction of electric field. This couple is called the torque and is denoted by τ. ∴ torque τ = magnitude of one force ´ perpendicular distance between lines of action of forces = qE (BN)
= qE (2lsinθ)
=(q 2l) Esinθ
= pEsinq [usinθ(1)] ....(2)
Clearly, the magnitude of torque depends on orientation (θ) of the electric dipole relative to electric field. Torque (τ) is a vector quantity whose direction is perpendicular to both vector pand vector E .
In vector form vector τ = vector p x vector E ...(3) Thus, if an electric dipole is placed in an electric field in oblique orientation, it experiences no force but experiences a torque. The torque tends to align the dipole moment along the direction of electric field.