Drive an expression for an electric dipole placed in a uniform electric field
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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) Force: The force on charge + q is, vector F1 = q vector E , along the direction of field vector E The force on charge - q is, vector F2 = qE , opposite to the direction of field vector E Obviously forces vector F1 and F2 are equal in magnitude but opposite in direction; hence net force on electric dipole in uniform electric field is F = F1 - F2 = qE - qE = 0 (zero) As net force on electric dipole is zero, so dipole does not undergo any translatory motion. Torque: 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 p and 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. Maximum Torque: For maximum torque sinθ should be the maximum. As the maximum value of sin q =1 when θ = 90° ∴ Maximum Torque, τmax = pE