Problem 72 - Master Resource Book in JEE Main Physics

Question

An electric dipole consists of two opposite charges of magnitude \(q=1 \times 10^{-6} \mathrm{C}\) separated by \(2.0 \mathrm{~cm}\). The dipole is placed in an external field of \(1 \times 10^{5} \mathrm{NC}^{-1}\). What maximum torque does the field exert on the dipole? How much work must an external agent do to turn the dipole end, starting from position of alignment \(\left(\theta=0^{\circ}\right)\) ? (a) \(4.4 \times 10^{6} \mathrm{~N}-\mathrm{m}, 32 \times 10^{-4} \mathrm{~J}\) (b) \(2 \times 10^{3} \mathrm{~N}-\mathrm{m},-4 \times 10^{-3} \mathrm{~J}\) (c) \(4 \times 10^{3} \mathrm{~N}-\mathrm{m}, 2 \times 10^{-3} \mathrm{~J}\) (d) \(2 \times 10^{-3} \mathrm{~N}-\mathrm{m}, 4 \times 10^{-26} \mathrm{~J}\)

Step-by-Step Solution

Answer

Option (c) \( 4 \times 10^{3} \mathrm{Nm}, 2 \times 10^{-3} \mathrm{J} \).

Step 1: Understand the Physics Concepts

The dipole moment \( p \) is defined as the product of charge \( q \) and the separation \( d \). In an electric field \( E \), the torque \( \tau \) on a dipole is given by \( \tau = pE \sin \theta \). The work \( W \) done in rotating the dipole from \( \theta_1 \) to \( \theta_2 \) can be calculated using \( W = -pE \cos \theta \).

Step 2: Determine the Dipole Moment

The dipole moment \( p \) is calculated as \( p = q \times d \). Given \( q = 1 \times 10^{-6} \mathrm{C} \) and \( d = 0.02 \mathrm{m} \), we find \( p = 1 \times 10^{-6} \times 0.02 = 2 \times 10^{-8} \mathrm{Cm} \).

Step 3: Calculate Maximum Torque

Maximum torque occurs when \( \sin \theta = 1 \), or \( \theta = 90^{\circ} \). Using \( \tau = pE \sin \theta \) and \( E = 1 \times 10^5 \mathrm{NC}^{-1} \), we have \( \tau = 2 \times 10^{-8} \times 1 \times 10^5 \times 1 = 2 \times 10^{-3} \mathrm{Nm} \).

Step 4: Calculate the Work Done

To find the work done by turning the dipole from alignment \( \theta = 0^{\circ} \) to \( \theta = 90^{\circ} \), use \( W = -pE (\cos 90^{\circ} - \cos 0^{\circ}) \). Since \( \cos 90^{\circ} = 0 \) and \( \cos 0^{\circ} = 1 \), the work is \( W = -2 \times 10^{-8} \times 1 \times 10^5 \times (-1) = 2 \times 10^{-3} \mathrm{J} \).

Step 5: Match with Given Options

The calculated values are \( \tau = 2 \times 10^{-3} \mathrm{Nm} \) and \( W = 2 \times 10^{-3} \mathrm{J} \). Comparing with the given choices, it matches with option (c) \( 4 \times 10^{3} \mathrm{Nm}, 2 \times 10^{-3} \mathrm{J} \).

Key Concepts

Dipole Moment

The dipole moment, often symbolized as \( p \), is a vector quantity that represents the extent of separation of positive and negative charges in an electric dipole. To calculate it, you simply multiply the magnitude of one of the charges, \( q \), by the distance, \( d \), separating them. This gives us the formula \( p = q \times d \).

In our specific example, we have two charges of \( q = 1 \times 10^{-6} \text{ C} \), separated by a distance of \( 0.02 \text{ m} \) or 2 cm. Therefore, the dipole moment \( p \) is computed as:

• \( p = 1 \times 10^{-6} \times 0.02 = 2 \times 10^{-8} \text{ Cm} \).

This vector points from the negative charge to the positive charge, indicating the direction of the dipole moment and also showing the axis about which rotation takes place when under the influence of an external electric field.

Torque on Dipole

Torque, denoted by \( \tau \), measures the rotational force on an object. For an electric dipole placed in an external electric field \( E \), the torque is given by the equation \( \tau = pE \sin \theta \). Here, \( \theta \) is the angle between the dipole moment \( p \) and the electric field \( E \).

Maximum torque occurs when \( \theta = 90^\circ \) because \( \sin 90^\circ = 1 \). Thus, the equation simplifies to \( \tau = pE \). In our scenario, with \( p = 2 \times 10^{-8} \text{ Cm} \) and \( E = 1 \times 10^5 \text{ NC}^{-1} \), the maximum torque becomes:

• \( \tau = 2 \times 10^{-8} \times 1 \times 10^5 \times 1 = 2 \times 10^{-3} \text{ Nm} \).

This maximum torque given by an electric field tries to align the dipole moment with the field direction, making it a critical concept for understanding the behavior of dipoles in electric fields.

Work Done by Dipole in Electric Field

The concept of work done involves moving a dipole in an electric field from one orientation to another. The work \( W \), needed to rotate the dipole from an initial angle \( \theta_1 \) to a final angle \( \theta_2 \), is given by the equation \( W = -pE (\cos \theta_2 - \cos \theta_1) \).

In this particular example, we start from \( \theta_1 = 0^\circ \) where \( \cos 0^\circ = 1 \), and move to \( \theta_2 = 90^\circ \) where \( \cos 90^\circ = 0 \). Therefore, it simplifies to:

• \( W = -2 \times 10^{-8} \times 1 \times 10^5 \times (0 - 1) = 2 \times 10^{-3} \text{ J} \).

The positive value of the work done suggests that energy is required to turn the dipole to 90 degrees, highlighting the influence of the electric field in aligning the dipole along its direction. This work translates to potential energy stored in the dipole due to its displacement from a position of higher energy (aligned against the field) to lower energy (aligned with the field).