An electric dipole consists of two opposite charges each of magnitude 1 µC, separated by 2 cm. The dipole is placed in an external electric field of \( 10^5 \, \text{N/C} \).
Calculate the:
(i) maximum torque experienced by the dipole
(ii) work done by the external field to turn the dipole through 180°
Show Hint
The torque on an electric dipole is maximum when the dipole is perpendicular to the electric field, and the work done to rotate the dipole is the change in potential energy.
Step 1: Formula for Torque.
The torque \( \tau \) experienced by an electric dipole in a uniform electric field is given by:
\[
\tau = pE \sin \theta
\]
where \( p \) is the dipole moment, \( E \) is the electric field, and \( \theta \) is the angle between the dipole moment and the electric field.
Step 2: Calculate Dipole Moment.
The dipole moment \( p \) is given by:
\[
p = q \times d
\]
where \( q = 1 \, \mu C = 1 \times 10^{-6} \, \text{C} \) is the charge and \( d = 2 \, \text{cm} = 0.02 \, \text{m} \) is the separation between the charges. Substituting the values:
\[
p = (1 \times 10^{-6}) \times 0.02 = 2 \times 10^{-8} \, \text{C·m}
\]
Step 3: Maximum Torque.
The maximum torque occurs when \( \sin \theta = 1 \), i.e., when \( \theta = 90^\circ \). The maximum torque is then:
\[
\tau_{\text{max}} = pE
\]
Substituting \( p = 2 \times 10^{-8} \, \text{C·m} \) and \( E = 10^5 \, \text{N/C} \):
\[
\tau_{\text{max}} = (2 \times 10^{-8}) \times (10^5) = 2 \times 10^{-3} \, \text{N·m}
\]
Step 4: Work Done.
The work done to rotate the dipole from \( \theta = 0^\circ \) to \( \theta = 180^\circ \) is given by the change in potential energy:
\[
W = pE \left( \cos \theta_2 - \cos \theta_1 \right)
\]
Substituting \( \theta_1 = 0^\circ \) and \( \theta_2 = 180^\circ \):
\[
W = pE \left( \cos 180^\circ - \cos 0^\circ \right)
\]
\[
W = pE (-1 - 1) = -2pE
\]
Substituting \( p = 2 \times 10^{-8} \, \text{C·m} \) and \( E = 10^5 \, \text{N/C} \):
\[
W = -2 \times (2 \times 10^{-8}) \times (10^5) = -4 \times 10^{-3} \, \text{J}
\]
Thus, the work done is \( 2 \, \text{J} \).
