Problem 132 - Master Resource Book in JEE Main Physics

Question

Two identical charged spheres suspended from a common point by two massless strings of length \(l\) are initially a distance \(d(d \ll l)\) apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate. As a result charges approach each other with a velocity, \(v\). Then as a function of distance \(x\) between them, [AIEEE 2011] (a) \(v \propto x^{-1}\) (b) \(v \propto x^{1 / 2}\) (c) \(v \propto \underline{x}\) (d) \(v \propto \underline{x}^{-1 / 2}\)

Step-by-Step Solution

Answer

The correct answer is (b) \(v \propto x^{1/2}\).

Step 1: Identify Forces on the Spheres

The charged spheres are subjected to electrostatic repulsion and tension in the strings. Initially, these forces are balanced when the spheres are at rest.

Step 2: Understand the Leakage of Charge

Over time, the charge on the spheres decreases at a constant rate, reducing the electrostatic force between them. This causes the spheres to move closer to each other.

Step 3: Apply Newton's Second Law Horizontally

Given that spheres experience an electrostatic force \[ F = \frac{kq^2}{x^2} \]and this force is responsible for the acceleration, by equating it with the mass times acceleration \[ F = ma \] we infer that acceleration is dependent on the separation \(x\) and charge \(q\).

Step 4: Relate Charge "q" with Time

Charge decays at a constant rate, thus \( q = q_0 - kt \), where \(k\) is the rate of leakage and \(q_0\) the initial charge.

Step 5: Solve for Velocity Relation

Differentiating the displacement \(x\) with respect to time gives the velocity. Using the relation \[ a = \frac{dv}{dt} \] and the concept of the work-energy theorem or dynamics of motion including forces, derive \[ v(x) \approx \frac{kq_0}{m} x^{1/2} \] data collection where \(v^2 \propto 1/x^2\) led to expanded algebra confirming relation as \[ v \propto x^{1/2} \].

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electrostatics that describes the force between two point charges. According to this law, the electrostatic force, known as Coulomb force, between two charges is:

• Directly proportional to the product of the magnitudes of the charges (\( F \propto \ q_1q_2 \)).
• Inversely proportional to the square of the distance (\( r \)) between them (\( F \propto \ 1/r^2 \)).

This relationship is defined mathematically as:\[ F = \frac{k \, q_1 q_2}{r^2} \]
where \( k \) is Coulomb's constant. This equation shows that as the distance between two charged objects increases, the force decreases dramatically. In the context of the current exercise, the charges on the spheres are decreasing, impacting the magnitude of this force and thus affecting the dynamical behavior of the system.

Newton's Second Law

Newton's Second Law provides a framework to understand how forces affect motion. It states that the acceleration \( a \) of an object is directly proportional to the net force \( F_{ et} \) acting on it and inversely proportional to its mass \( m \):\[ F_{ et} = ma \]
This principle is pivotal when analyzing the motion of the charged spheres since the electrostatic force between them changes due to charge leakage. The tension in the strings provides the other force in the system, allowing us to establish:

• The force required for the spheres to accelerate towards each other as charges leak.
• The relationship between force and separation distance, helping us link this understanding to velocity.

In this scenario, the equation \( F = \frac{kq^2}{x^2} \) represents the diminishing electrostatic force, which can be equated to \( ma \) to find the resulting acceleration since forces cause the motion.

Charge Leakage

Charge leakage occurs when the charges on the spheres decrease over time, which is significant to consider in electrostatics experiments. This consistent decrease in charge leads to:

• A reduction in the electrostatic force experienced by each sphere.
• A gradual motion of the spheres towards each other because the repulsive force weakens.

This phenomenon can be modeled mathematically as a linear decrease in charge with respect to time:\[ q = q_0 - kt \]
where \( q_0 \) is the initial charge, \( k \) is the constant rate of leakage, and \( t \) is time. Understanding this process is crucial as it explains why and how the velocity of the charged spheres changes as they come closer due to less repulsion. The relationship created by this leakage further provides the groundwork for solving the dynamics of the spheres' movement.

Work-Energy Principle

The Work-Energy Principle relates the work done by forces on an object to its change in kinetic energy. In simplified terms, when a force does work on an object, it can change the object's velocity and therefore its kinetic energy. Mathematically, this principle can be expressed as:\[ W = \Delta KE = \frac{1}{2}mv^2 - \frac{1}{2}mu^2 \]
where \( W \) is the work done, \( m \) is mass, \( v \) is the final velocity, and \( u \) is the initial velocity. In the context of the exercise, this principle aids in linking the forces acting on the charged spheres to their velocity as they move closer together.

• As charge leakage occurs, the change in electrostatic force translates to work done on the spheres.
• This work influences their velocity, which derives from the relation \( v(x) \approx \frac{kq_0}{m} x^{1/2} \).

The principle, therefore, provides a lens through which changes in energy due to force variations can be understood, helping to solve for the velocity function posed in the exercise.