Question: 35. An electric dipole with dipole moment 5 \times 10^{-6} \mathrm{Cm} is aligned with the direction of a uniform electric field of magnitude 4 \times 10^{5} \mathrm{~N} / \mathrm{C}. The dipole is then rotated through an angle of 60^{\circ} with respect to the electric field. The change in the potential energy of the dipole is:
(1) 0.8 J
(2) 1.0 J
(3) 1.2 J
(4) 1.5 J
Answer: Option (2)
Explanation:
The potential energy U of an electric dipole placed in a uniform electric field is given by:
U = -pE \cos \thetawhere p is the dipole moment, E is the electric field,
and \theta is the angle between the dipole moment and the electric field.
Initially, the dipole is aligned with the electric field, so:
\theta_{1} = 0^\circInitial potential energy:
U_{1} = -pE \cos 0^\circ = -pEAfter rotation, the angle between the dipole and the field becomes:
\theta_{2} = 60^\circFinal potential energy:
U_{2} = -pE \cos 60^\circThe change in potential energy is:
\Delta U = U_{2} - U_{1}Substituting the expressions:
\Delta U = -pE \cos 60^\circ + pE \Delta U = pE (1 - \cos 60^\circ)Now substitute the given values:
p = 5 \times 10^{-6} \ \mathrm{Cm} E = 4 \times 10^{5} \ \mathrm{N/C} pE = 5 \times 10^{-6} \times 4 \times 10^{5} = 2 \ \mathrm{J}Since \cos 60^\circ = \frac{1}{2},
\Delta U = 2 \left(1 - \frac{1}{2}\right) = 1 \ \mathrm{J}Therefore, the change in potential energy of the dipole is 1.0 J.
Hence, the correct answer is Option (2).