Problem 83 - Master Resource Book in JEE Main Physics

Question

A charge \((-q)\) and another charge \((+Q)\) are kept a two points \(A\) and \(B\), respectively. Keeping the charg \((+Q)\) fixed at \(B\), the charge \((-q)\) at \(A\) is moved \(t\) another point \(C\) such that \(A B C\) forms an equilatera triangle of side \(l\). The net work done in moving th charge \((-q)\) is (a) \(\frac{1}{4 \pi \varepsilon_{0}} \frac{Q q}{l}\) (b) \(\frac{1}{4 \pi \varepsilon_{0}} \frac{Q q}{l^{2}}\) (c) \(\frac{1}{4 \pi \varepsilon_{0}} Q q l\) (d) zero

Step-by-Step Solution

Answer

The net work done is zero (Option d).

Step 1: Understand the Initial and Final Positions of Charge

Initially, charge (-q) is at point A and we are moving it to point C such that points A, B, and C form an equilateral triangle with side length l. Charge (+Q) is kept fixed at point B.

Step 2: Calculate Initial Potential Energy

The potential energy of the system when charge (-q) is at point A relative to charge (+Q) at point B is given by \( U_i = \frac{1}{4\pi\varepsilon_0} \cdot \frac{-qQ}{l} \).

Step 3: Calculate Final Potential Energy

The potential energy of the system when charge (-q) is moved to point C is the same as the initial situation because the distances are equal in an equilateral triangle, hence \( U_f = \frac{1}{4\pi\varepsilon_0} \cdot \frac{-qQ}{l} \).

Step 4: Calculate the Net Work Done

The work done (W) in moving the charge is the difference in potential energy from initial to final positions. Thus, \( W = U_f - U_i = 0 \). This indicates that there's no net work done since the initial and final energies are the same.

Key Concepts

Potential Energy

Potential energy in electrostatics refers to the energy stored due to the position of a charge in an electric field. When a charge is placed near another charge, it experiences an electric force. This force has the potential to do work if the charge moves.

The potential energy of a system of two point charges can be calculated using the formula:

• Initial potential energy, when the charge (-q) is at point A relative to charge (+Q) at point B, is given by:\[ U_i = \frac{1}{4\pi\varepsilon_0} \cdot \frac{-qQ}{l} \]

The potential energy depends on:

• The magnitude of the charges (q) and (Q).
• The distance (l) between them.
• The Coulomb's constant (\frac{1}{4\pi\varepsilon_0})

Potential energy indicates how much work would be required to bring the charge from infinity to the point of interest under the influence of the electric field.

Work Done

In physics, work is done when a force remains applied to an object and causes a displacement. In the context of electrostatics, work done is closely related to changes in potential energy.

For moving a charge from one position to another, the work done is equal to the change in potential energy. This is expressed as:

• \( W = U_f - U_i \)

When the charge (-q) is moved from point A to point C in the given equilateral triangle setup, we find:

• Initial potential energy, (U_i), is equal to the final potential energy, (U_f).
• This means no net work is done, so \( W = 0 \).

Since both positions yield the same potential energy due to the symmetry of the equilateral triangle, the work done is zero.

Equilateral Triangle

An equilateral triangle is a polygon with three equal-length sides and all angles equal to 60 degrees.

In the context of electrostatics, geometries such as equilateral triangles provide a symmetric arrangement, simplifying calculations.
In our example:

• The triangle has sides labeled (A, B, C) , each of length (l) .
• This symmetry ensures equal distances among all vertices, implying equal potential energies if charges are placed at these points.

When we move a charge in such a well-defined geometric setup, the outcome in terms of work and energy is determined fundamentally by this symmetry.

Coulomb's Law

Coulomb's Law is a fundamental principle of electrostatics that defines the force between two point charges. The equation for Coulomb's Law is given by:

• \( F = \frac{1}{4\pi\varepsilon_0} \cdot \frac{|q_1 q_2|}{r^2} \)

where:

• \( F \) is the magnitude of force between the charges.
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges.
• \( r \) is the distance between the centers of two charges.
• \( \frac{1}{4\pi\varepsilon_0} \) is the proportionality constant.

In our scenario, using Coulomb’s Law helps in understanding the forces between charges and consequently their potential energies.

Understanding this law allows us to calculate and predict interactions within any electric field system.