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

The question involves calculating the work done on an electric dipole when it is rotated from its equilibrium position through \( 180^\circ \) in a uniform electric field. The key concepts here are the electric dipole moment and the potential energy in an external electric field.

Conceptual Approach: 
An electric dipole consists of two equal and opposite charges separated by a distance. When placed in a uniform electric field, the dipole experiences torque which tries to align it along the field direction. The work done on the dipole when rotated in the electric field is given by changing its potential energy.

Formulas Required:

• The dipole moment \( \mathbf{p} \) is given by: \(\mathbf{p} = q \cdot \mathbf{d}\), where \( q \) is the charge and \( \mathbf{d} \) is the distance vector between the charges.
• The potential energy of a dipole in an electric field \( \mathbf{E} \) is: \(U = -\mathbf{p} \cdot \mathbf{E} = -pE \cos \theta\), where \( \theta \) is the angle between \( \mathbf{p} \) and \( \mathbf{E} \).
• Work done \( W \) when rotating the dipole from \( \theta_1 \) to \( \theta_2 \) is: \(W = U(\theta_2) - U(\theta_1)\).

 

Calculation:

1. Calculate the dipole moment \( \mathbf{p} \):
   The charges are \( -4 \, \mu \text{C} \) and \( 4 \, \mu \text{C} \), and the distance between them is from \( (-9, 0, 0) \) to \( (9, 0, 0) \), which is 18 cm or 0.18 m.
   \(p = q \cdot d = 4 \times 10^{-6} \, \text{C} \times 0.18 \, \text{m} = 7.2 \times 10^{-7} \, \text{Cm}\)
2. Calculate Initial Potential Energy \( U(\theta_1 = 0^\circ) \):
   \(U(0^\circ) = -pE \cos(0^\circ) = -7.2 \times 10^{-7} \times 10^4 \times 1 = -7.2 \times 10^{-3} \, \text{J}\)
3. Calculate Final Potential Energy \( U(\theta_2 = 180^\circ) \):
   \(U(180^\circ) = -pE \cos(180^\circ) = -7.2 \times 10^{-7} \times 10^4 \times (-1) = 7.2 \times 10^{-3} \, \text{J}\)
4. Calculate the Work Done:
   \(W = U(180^\circ) - U(0^\circ) = 7.2 \times 10^{-3} - (-7.2 \times 10^{-3}) = 14.4 \times 10^{-3} \, \text{J} = 14.4 \, \text{mJ}\)

Therefore, the work done on the dipole when rotated by \( 180^\circ \) is 14.4 mJ.

Answer 2

The work done on a dipole in an electric field when rotated through an angle \( \theta \) is given by: \[ W = -pE \cos(\theta_2) + pE \cos(\theta_1), \] where \( p \) is the dipole moment, \( E \) is the electric field strength, and \( \theta_1 = 0^\circ \) and \( \theta_2 = 180^\circ \). The dipole moment \( p \) is given by: \[ p = q \cdot d, \] where \( q = 4 \, \mu C = 4 \times 10^{-6} \, \text{C} \) and \( d = 9 \, \text{cm} = 0.09 \, \text{m} \). Thus: \[ p = 4 \times 10^{-6} \cdot 0.18 = 7.2 \times 10^{-7} \, \text{C} \cdot \text{m}. \] Now, calculate the work done: \[ W = - (7.2 \times 10^{-7} \cdot 10^4) \left( \cos(180^\circ) - \cos(0^\circ) \right) = - (7.2 \times 10^{-7} \cdot 10^4) \left( -1 - 1 \right). \] \[ W = 14.4 \times 10^{-3} = 14.4 \, \text{mJ}. \]