Problem 20 - Master Resource Book in JEE Main Physics

Question

An electric dipole consists of two opposite charges, each of magnitude \(1.0 \mu \mathrm{C}\) separated by a distance of \(2.0 \mathrm{~cm}\). The dipole is placed in an external electric field of \(10^{5} \mathrm{NC}^{-1}\). The maximum torque on the dipole is (a) \(0.2 \times 10^{-3} \mathrm{~N}-\mathrm{m}\) (b) \(1 \times 10^{-3} \mathrm{~N}-\mathrm{m}\) (c) \(2 \times 10^{-3} \mathrm{~N}-\mathrm{m}\) (d) \(4 \times 10^{-3} \mathrm{~N}-\mathrm{m}\)

Step-by-Step Solution

Answer

(c) \(2 \times 10^{-3} \mathrm{~N}-\mathrm{m}\).

Step 1: Understand the Problem

We are given an electric dipole consisting of two charges, each with a magnitude of \(1.0 \mu C\), separated by a distance of \(2.0\, \text{cm}\). The dipole is in an external electric field of \(10^5\, \text{N/C}\). Our task is to find the maximum torque on the dipole when it is placed in this electric field.

Step 2: Recall the Torque Formula

The torque \(\tau\) experienced by an electric dipole in an electric field is given by the formula: \[ \tau = pE \sin(\theta) \]where \(p\) is the dipole moment, \(E\) is the magnitude of the electric field, and \(\theta\) is the angle between the dipole moment and the electric field. The maximum torque occurs when \(\sin(\theta) = 1\); that is, \(\theta = 90^\circ\). So, the maximum torque is \(\tau_{\text{max}} = pE\).

Step 3: Calculate the Dipole Moment

The dipole moment \(p\) is calculated as the product of the charge magnitude \(q\) and the separation distance \(d\) (converted to meters):\[ p = q \times d = (1.0 \times 10^{-6} \text{ C}) \times (2.0 \times 10^{-2} \text{ m}) = 2.0 \times 10^{-8} \text{ C} \cdot \text{m} \]

Step 4: Calculate the Maximum Torque

Substitute the calculated dipole moment \(p\) and the given electric field \(E\) into the torque formula to find the maximum torque:\[ \tau_{\text{max}} = pE = (2.0 \times 10^{-8} \text{ C} \cdot \text{m}) \times (10^5 \text{ N/C}) = 2.0 \times 10^{-3} \text{ N} \cdot \text{m} \]

Step 5: Choose the Correct Answer

We found that the maximum torque \(\tau_{\text{max}}\) is \(2.0 \times 10^{-3} \text{ N} \cdot \text{m}\). Comparing this with the provided options, we find that option (c) matches our calculation.

Key Concepts

Torque on Dipole

When it comes to understanding how an electric dipole behaves in an electric field, torque is a key concept. Torque, in physics, is a measure of the force that can cause an object to rotate about an axis. For an electric dipole situated in an electric field, this rotation or turning effect is paramount. The formula that represents torque (\( \tau \)) on a dipole is given by:\[\tau = pE \sin(\theta)\]Where:

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

The torque will be at its maximum when the angle \( \theta \) is 90 degrees (\( \sin(\theta) = 1 \)), meaning the dipole is perpendicular to the electric field. This configuration maximizes the rotational effect of the torque. In situations involving dipoles in fields, asking about the maximum torque is common, as it provides insights into the potential alignment and energy characteristics of the dipole system.

Dipole Moment

The dipole moment is a vital concept that describes the separation of positive and negative charges within a dipole. It is a vector quantity, meaning it has both magnitude and direction. The magnitude of the dipole moment (\( p \)) is calculated from the product of the charge magnitude (\( q \)) and the separation distance (\( d \)) between the charges. The formula to compute the dipole moment is:\[p = q \times d\]For instance, taking charges of magnitude \( 1.0 \mu C \) separated by \( 2.0 \, \text{cm} \) means the dipole moment would be:\[p = (1.0 \times 10^{-6} \text{ C}) \times (2.0 \times 10^{-2} \text{ m}) = 2.0 \times 10^{-8} \text{ C} \cdot \text{m}\]The dipole moment points from the negative charge to the positive charge, offering a directional attribute to the concept. Understanding the dipole moment is crucial because it provides foundational insights into the interaction of dipoles with external electric fields which influence their orientation and behavior.

Electric Field

Electric fields are fundamental to the understanding of electrostatic forces. An electric field (\( E \)) is a region around a charged particle where a force is exerted on other charged objects. It is represented as a vector, showing both the magnitude and direction of the force it exerts upon charges.#### Characteristics of Electric Fields:

• Direction: Points from a positive charge to a negative charge.
• Magnitude: Determined by the charge creating it and the distance from the charge.

For a uniform electric field, often encountered in problems involving dipoles, the field strength is constant both in magnitude and direction. In our example, the electric field (\( 10^5 \, \text{N/C} \)) affects the electric dipole by exerting forces on its charges, leading to torques that attempt to align the dipole along the field direction.Understanding electric fields is key when analyzing how dipoles interact within these fields, as they dictate the forces and torques experienced, ultimately affecting the dipole's orientation and energy.