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

To find the change in potential energy of the dipole when it is rotated in an electric field, we use the formula for the potential energy \( U \) of a dipole in an electric field:

\[ U = -pE\cos\theta \]

where:

• \( p \) is the dipole moment,
• \( E \) is the electric field strength,
• \( \theta \) is the angle between the dipole moment and the electric field.

Initially, the dipole is aligned with the electric field, so the initial angle \( \theta_1 = 0^\circ \). The potential energy is:

\[ U_1 = -pE\cos(0^\circ) = -pE \]

After rotation by \( 60^\circ \), the new angle \( \theta_2 = 60^\circ \), and the potential energy becomes:

\[ U_2 = -pE\cos(60^\circ) \]

The change in potential energy \(\Delta U\) is given by:

\[ \Delta U = U_2 - U_1 \]

Substitute the expressions for \( U_1 \) and \( U_2 \):

\[ \Delta U = -pE\cos(60^\circ) - (-pE) = pE(1 - \cos(60^\circ)) \]

Given \(\cos(60^\circ) = 0.5\), the expression becomes:

\[ \Delta U = pE(1 - 0.5) = pE \cdot 0.5 \]

Substitute \( p = 5 \times 10^{-6} \, \text{Cm} \) and \( E = 4 \times 10^5 \, \text{N/C} \):

\[ \Delta U = 5 \times 10^{-6} \times 4 \times 10^5 \times 0.5 = 1 \, \text{J} \]

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