Problem 3 - Master Resource Book in JEE Main Physics

Question

Two point charges repel each other with a force of \(100 \mathrm{~N}\). One of the charges is increased by \(10 \%\) and other is reduced by \(10 \%\). The new force of repulsion at the same distance would be (a) \(100 \mathrm{~N}\) (b) \(121 \mathrm{~N}\) (c) \(99 \mathrm{~N}\) (d) None of these

Step-by-Step Solution

Answer

The new force is 99 N, so the correct option is (c).

Step 1: Understand the Original Scenario

The initial force of repulsion between two point charges is given as \( F_0 = 100 \) N. Coulomb's Law tells us that this force is proportional to the product of the magnitudes of the two charges, i.e., \( F_0 = k \frac{{q_1 \cdot q_2}}{{r^2}} \), where \( k \) is Coulomb's constant and \( r \) is the distance between the charges.

Step 2: Adjust the Charges

One charge increases by \(10\%\), so the new charge \( q_1' = 1.1q_1 \). The other charge decreases by \(10\%\), so the new charge \( q_2' = 0.9q_2 \).

Step 3: Define the New Force

The new force, \( F' \), using Coulomb's law is described by \( F' = k \frac{{q_1' \cdot q_2'}}{{r^2}} = k \frac{{1.1q_1 \cdot 0.9q_2}}{{r^2}} \).

Step 4: Express the New Force Relative to the Original Force

Substitute the adjusted charges into the expression for the force: \( F' = k \frac{{1.1q_1 \cdot 0.9q_2}}{{r^2}} = 1.1 \times 0.9 \times k \frac{{q_1 \cdot q_2}}{{r^2}} = 0.99 \times k \frac{{q_1 \cdot q_2}}{{r^2}} \).

Step 5: Calculate the New Force

Since \( k \frac{{q_1 \cdot q_2}}{{r^2}} = 100 \) N from the original setup, then \( F' = 0.99 \times 100 = 99 \) N.

Step 6: Determine the Correct Option

The new force is calculated as 99 N. Therefore, the correct answer from the given options is (c) \(99 \mathrm{~N}\).

Key Concepts

Electrostatic Force

Electrostatic force is a fundamental concept in physics that explains the interaction between charged objects. It is described by Coulomb's Law, which provides a formula to calculate the magnitude of this force between two point charges. The formula is represented as \(F = k \frac{q_1 \cdot q_2}{r^2},\) where \(F\) is the electrostatic force, \(q_1\) and \(q_2\) are the magnitudes of the charges, \(r\) is the distance between them, and \(k\) is Coulomb's constant.
This force can be either attractive or repulsive, depending on the nature of the charges. Like charges repel each other, while opposite charges attract.

• Repulsive force: Occurs between charges with the same sign.
• Attractive force: Occurs between charges with opposite signs.

Understanding the electrostatic force is crucial for solving many physics problems relating to charge interactions.

Point Charges

Point charges are idealized charges that are assumed to be located at a single point in space. This simplification allows for easier calculations and theoretical analysis, especially when using Coulomb's Law to determine forces between them. In reality, charges are distributed over volumes, but in many cases, treating them as point charges provides sufficient accuracy for the problem at hand.
In the context of the given exercise, we are dealing with two point charges. The use of point charges helps in simplifying and focusing on the crucial aspect of the problem, which is the interaction between the charges. There's no need to worry about the distribution of the charge over a larger area or volume.

• Assumed to have negligible size.
• Helpful in simplifying physics problems involving charge interactions.

This concept is fundamental to understanding many phenomena in electromagnetism.

Charge Interaction

Charge interaction refers to how different electrical charges influence each other when in proximity. The magnitude and direction of the interaction are dictated by Coulomb's Law. Sometimes, charges are static, while in other times, they may move, resulting in dynamic interactions.
In our exercise, the interaction is repulsive because both charges are similar in nature (both positive or both negative). When the charges are modified - one is increased and the other decreased - the interaction magnitude changes accordingly. Specifically, if one charge is increased by \(10\%\) and the other reduced by \(10\%\), it leads to a change in the overall force by \(1\%\).

• Charge increase: Amplifies the interaction force.
• Charge decrease: Weakens the interaction force.

Recognizing how charge interactions vary is key to correctly applying Coulomb's Law in physics problems.

Physics Problem Solving

Physics problem solving involves breaking down complex scenarios into comprehensible parts using established principles and logical reasoning.
In our given problem, we follow a systematic approach:

• Identify the Given Information: The initial force and percentage changes in charge.
• Apply the Right Laws: Use Coulomb's Law to relate charges and forces.
• Calculate Step-by-Step: Account for the percentage increase and decrease in charges to find the new force of repulsion.

Using these steps simplifies the problem-solving process. It ensures clarity and accuracy in deriving the solution. Developing a structured approach to physics problems not only makes them easier but also enhances one's ability to tackle more complex scenarios in the future.