Question

(a) Derive an expression for the torque experienced by an electric dipole kept in a uniformly electric field. 

(b) Calculate the work done to dissociate the system of three charges placed on the vertices of a triangle as shown. Here q =1.6 x10^-10C.

Answer 1

(a) 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 = qvector 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 F and vector F 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 x perpendicular distance between lines of action of forces 

= qE(BN) = qE(2lsinθ) 

=(θ2l)Esinθ 

= pEsinθ[using (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 sinθ =1 when θ = 90° 

∴ Maximum Torque,τ​_max = pE