A dipole of moment 0.1 e · nm is placed in a uniform electric field with a magni

Question

A dipole of moment 0.1 e · nm is placed in a uniform electric field with a magnitude of 30000 N/C.

a. What is the magnitude of the torque on the dipole when the dipole is parallel to the electric field? The fundamental charge is 1.602 × 1019 C. Answer in units of Nm.

b. What is the magnitude of the torque on the dipole when the dipole is perpendicular to the electric field? Answer in units of Nm.

c. What is the magnitude of the torque on the dipole when the dipole makes an angle of 20 with the electric field? Answer in units of Nm.

d. Find the potential energy of the dipole in the electric field at 20 . Answer in units of J.

Explanation / Answer

According to the given problem,

a)  0 N-m

Reason: There is no component of the electric field perpendicular to the charges, and a perpendicular force is necessary to generate torque.

b) Concept: Notice that when the dipole is orthogonal to the field, the electric force on the positive and negative ends of the dipole are opposite one another. However, the two ends are also on opposite sides of the center of mass of the dipole, and consequently the two torques act to rotate the dipole in the same direction: hence, the torques will add. For rotation about the center of mass d/2 (d being the separation distance of the dipole), the electric field produces a force of magnitude qE on the upper half, and so the torque is rF = (d/2)qE. Similarly, there is another torque (d/2)qE on the lower half.

Consequently the net torque on the dipole is

dqE = pE = (0.1 e nm)(30000 N/C) = 0.1*10-9*1.602*10-19*30000 = 4.806*10-25N-m

The forumla for torque on a dipole is frequently just given as = p x E = pE sin , but the explanation above is where that formula comes from.

c) Same analysis as (b), except that since the dipole is at an angle of 20° with the field, the field has component E sin 20° perpendicular to the dipole and E cos 20° parallel to the dipole. Hence, the electric force and thus torque on the dipole are equal to the answer to

T = (b) times the sin20° = 1.644*10-25 N-m.

d) Just as the torque on a dipole is equal to p x E = pE sin , the potential energy of a dipole is given by U = - p E = - pE cos .

In this case we have (0.01 e nm)(30000 N/C)cos20° = 4.52*10-26 J.

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