Question

An electric dipole with dipole moment \(p = 5 \times 10^{-6}\) Cm is aligned with the direction of a uniform electric field of magnitude \(E = 4 \times 10^5\) N/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:

• A. \( 1.0 \text{ J} \)
• B. \( 1.2 \text{ J} \)
• C. \( 1.5 \text{ J} \)
• D. \( 0.8 \text{ J} \)

Hint:

Remember the formula for the potential energy of a dipole in an electric field \(U = -pE \cos \theta\). The change in potential energy is the difference between the final and initial potential energies.

Answer 1

The Correct Option is A

The change in potential energy of a dipole rotated in an electric field is determined using the potential energy formula \( U = -pE\cos\theta \), where \( p \) is the dipole moment, \( E \) is the electric field strength, and \( \theta \) is the angle between the dipole moment and the electric field.

Initially, the dipole is aligned with the field, so \( \theta_1 = 0^\circ \), resulting in an initial potential energy of \( U_1 = -pE\cos(0^\circ) = -pE \).

After rotation to \( \theta_2 = 60^\circ \), the new potential energy is \( U_2 = -pE\cos(60^\circ) \).

The change in potential energy, \( \Delta U \), is calculated as \( \Delta U = U_2 - U_1 \).

Substituting the expressions for \( U_1 \) and \( U_2 \) yields \( \Delta U = -pE\cos(60^\circ) - (-pE) = pE(1 - \cos(60^\circ)) \).

With \( \cos(60^\circ) = 0.5 \), the expression simplifies to \( \Delta U = pE(1 - 0.5) = 0.5pE \).

Using the given values \( p = 5 \times 10^{-6} \, \text{Cm} \) and \( E = 4 \times 10^5 \, \text{N/C} \), the change in potential energy is \( \Delta U = 5 \times 10^{-6} \times 4 \times 10^5 \times 0.5 = 1 \, \text{J} \).

Therefore, the change in potential energy of the dipole is \( 1.0 \, \text{J} \).