Problem 126 - Master Resource Book in JEE Main Physics

Question

Assertion-Reason type. Each of these contains two Statements : Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choice, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given below (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion (b) If both Assertion and Reason are true but Reason is not the correct explanation of the Assertion (c) If Assertion is true but Reason is false (d) If Assertion is false but the Reason is true Assertion An electric dipole is placed in a uniform electric field. Its equilibrium will be stable when dipole is set along the direction of electric field. Reason In stable equilibrium energy of dipole should be least possible.

Step-by-Step Solution

Answer

Choice (a): Both assertion and reason are true, and the reason is the correct explanation.

Step 1: Understanding the Assertion

The assertion states that an electric dipole in a uniform electric field reaches stable equilibrium when it aligns with the electric field. In this position, the torque on the dipole is zero, and it is considered to be in its lowest energy state.

Step 2: Understanding the Reason

The reason provided is that for an equilibrium to be stable, the energy of the system must be at its minimum. In stable equilibrium, the forces or torques are balanced, leading to a minimal energy configuration.

Step 3: Comparing Assertion with Reason

Both the assertion and the reason are true. An electric dipole aligns with the electric field to minimize its potential energy, which is at its lowest when the dipole is parallel to the field. This configuration corresponds to stable equilibrium.

Step 4: Determining if Reason Explains Assertion

Since the reason correctly explains why the dipole is stable in this orientation—because the energy is minimized—this reason is indeed the correct explanation of the assertion.

Step 5: Selecting the Correct Answer Choice

Based on the analysis, the correct choice is (a): Both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concepts

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a small distance. This system forms a pair of charges called dipole, represented by the equation \[\vec{p} = q \cdot \vec{d}\], where \( q \) is the charge of each particle and \( \vec{d} \) is the separation vector. This vector is directed from the negative to the positive charge.
Electric dipoles naturally occur in molecules, which have regions of positive and negative charge. They are important in fields such as electromagnetism because they can interact with electric fields, allowing us to study molecular properties and behaviors. Choosing a dipole moment can describe the overall polarity of a molecule, which influences how it behaves in various environments. Dipoles are foundational concepts in physics, especially in understanding interactions with electric fields and the forces that arise due to these interactions. Understanding electric dipoles helps explain why molecular compounds dissolve well in polar solvents like water.

Uniform Electric Field

A uniform electric field is one where the field strength is constant at every point. This field usually occurs between two parallel conductive plates with opposite charges, often used in physics experiments for their consistent behavior.
When thinking about a uniform electric field, visualize arrows pointing in the same direction and equally spaced. The strength and direction do not change, simplifying calculations involving forces and movements within this field.
For instance, when an electric dipole is placed in a uniform electric field, its behavior becomes predictable. The forces acting on it, due to a uniform field, lead to straightforward analysis, making it easier to predict the dipole's motion and energy changes. In real-life applications, uniform electric fields are crucial in devices like capacitors, where they store energy and regulate the flow of electric charge.

Stable Equilibrium

Stable equilibrium in physics implies a condition where, if a system is slightly disturbed, it will return to its original state. This concept is essential in understanding the behavior of systems under external influences.
For an electric dipole in a uniform electric field, stable equilibrium occurs when the dipole aligns with the electric field. In this position, any minor disturbance will not cause the dipole to remain deflected from its aligned position permanently. Instead, after the disturbance, it will return to being parallel to the field, signifying stability.
This concept extends to various systems in physics and engineering, where it is vital for ensuring that structures and systems can withstand small perturbations without catastrophic failures. Stable equilibrium is a sign of both physical stability and efficient energy use in systems.

Minimum Energy

When discussing systems in equilibrium, understanding the concept of minimum energy is crucial. A system at stable equilibrium is often in its state of lowest potential energy, meaning any deviation will increase its energy.
For an electric dipole, the minimum energy configuration is when it aligns parallel to an electric field. In this alignment, the potential energy is at its lowest, explaining why the system tends toward this position in a uniform electric field.
The principle of minimum energy helps explain natural phenomena where systems tend towards equilibrium states. In chemistry, this explains why reactions release energy as products form more stable configurations. In physics, this principle helps predict the behavior of physical systems, ensuring they operate in the most energy-efficient manner possible.

Torque in Electric Fields

Torque is a measure of the rotational force applied to an object. In an electric field, an electric dipole experiences a torque that tends to align it with the field lines. This interaction can be understood with the equation \[\tau = \vec{p} \times \vec{E}\], where \( \tau \) is the torque, \( \vec{p} \) is the dipole moment, and \( \vec{E} \) is the electric field strength.
The cross-product in the equation highlights that the torque is perpendicular to both the dipole moment and the electric field. This force tries to rotate the dipole toward a position where it will have zero torque once aligned. This alignment - parallel to the field - results in stable equilibrium and minimum energy.
Torque's role in electric fields is central in many technologies, such as electric motors, where torque is used to turn the motor, transforming electrical energy into mechanical motion. Understanding torque in the context of electric fields allows for innovations and improvements in electrical device designs.