Question

Two point charges \( -4 \, \mu C \) and \( 4 \, \mu C \), constituting an electric dipole, are placed at \( (-9, 0, 0) \, \text{cm} \) and \( (9, 0, 0) \, \text{cm} \) in a uniform electric field of strength \( 10^4 \, \text{N/C} \). The work done on the dipole in rotating it from the equilibrium through \( 180^\circ \) is:

• A. 14.4 mJ
• B. 18.4 mJ
• C. 12.4 mJ
• D. 16.4 mJ

Hint:

The work done in rotating a dipole in an electric field depends on the initial and final angles of the dipole's orientation.

Answer 1

The Correct Option is A

This problem requires calculating the work performed on an electric dipole rotated 180 degrees from its equilibrium position within a uniform electric field. Key concepts include the electric dipole moment and potential energy in an external electric field.

Conceptual Approach:
An electric dipole comprises two equal and opposite charges separated by a distance. In a uniform electric field, it experiences a torque that attempts to align it with the field. The work done on the dipole during rotation equals the change in its potential energy.

Formulas Required:

• Dipole moment \( \mathbf{p} \) is defined as: \( \mathbf{p} = q \cdot \mathbf{d} \), where \( q \) is the charge and \( \mathbf{d} \) is the vector representing the separation between the charges.
• The potential energy \( U \) of a dipole in an electric field \( \mathbf{E} \) is: \( U = -\mathbf{p} \cdot \mathbf{E} = -pE \cos \theta \), with \( \theta \) being the angle between \( \mathbf{p} \) and \( \mathbf{E} \).
• Work done \( W \) in rotating the dipole from an initial angle \( \theta_1 \) to a final angle \( \theta_2 \) is: \( W = U(\theta_2) - U(\theta_1) \).

 

Calculation:

1. Dipole moment \( \mathbf{p} \) calculation:
   Given charges of \( -4 \, \mu \text{C} \) and \( 4 \, \mu \text{C} \), and a separation distance of 18 cm (0.18 m) between \( (-9, 0, 0) \) and \( (9, 0, 0) \).
   \(p = q \cdot d = 4 \times 10^{-6} \, \text{C} \times 0.18 \, \text{m} = 7.2 \times 10^{-7} \, \text{Cm}\)
2. Initial Potential Energy \( U(\theta_1 = 0^\circ) \) calculation:
   \(U(0^\circ) = -pE \cos(0^\circ) = -7.2 \times 10^{-7} \times 10^4 \times 1 = -7.2 \times 10^{-3} \, \text{J}\)
3. Final Potential Energy \( U(\theta_2 = 180^\circ) \) calculation:
   \(U(180^\circ) = -pE \cos(180^\circ) = -7.2 \times 10^{-7} \times 10^4 \times (-1) = 7.2 \times 10^{-3} \, \text{J}\)
4. Work Done calculation:
   \(W = U(180^\circ) - U(0^\circ) = 7.2 \times 10^{-3} - (-7.2 \times 10^{-3}) = 14.4 \times 10^{-3} \, \text{J} = 14.4 \, \text{mJ}\)

The work done on the dipole for a 180° rotation is 14.4 mJ.