Question

Iwo small charged spheres repel each other with a force \(2 \times 10^{-3} \mathrm{~N}\). The charge on one sphere is twice that of the other. When these two spheres displaced \(10 \mathrm{~cm}\) further apart the force is \(5 \times 10^{-4} \mathrm{~N}\), then the charges on both the spheres are....... (A) \(1.6 \times 10^{-9} \mathrm{C}, 3.2 \times 10^{-9} \mathrm{C}\) (B) \(3.4 \times 10^{-9} \mathrm{C}, 11.56 \times 10^{-9} \mathrm{C}\) (C) \(33.33 \times 10^{9} \mathrm{C}, 66.66 \times 10^{-9} \mathrm{C}\) (D) \(2.1 \times 10^{-9} \mathrm{C}, 4.41 \times 10^{-9} \mathrm{C}\)

Short answer

The charges on both spheres are: (A) \(1.6 \times 10^{-9} \mathrm{C}\), \(3.2 \times 10^{-9} \mathrm{C}\).

Step-by-step solution

Understand the given information

Two charged spheres repel each other with an initial force of \(2 \times 10^{-3} \mathrm{N}\). The charge on one sphere is twice that of the other, so let's denote the charges as \(q_1\) and \(q_2\), where \(q_1 = 2q_2\). When the spheres are displaced \(10 \mathrm{~cm}\) further apart, the new force between them is \(5 \times 10^{-4} \mathrm{N}\).

Write the formula for Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged objects, and the formula is given by: \[F = k\frac{q_1q_2}{r^2}\] where \(F\) is the force between the charges, \(q_1\) and \(q_2\) are the charges, \(r\) is the distance between them, and \(k\) is Coulomb's constant (\(k = 8.988 \times 10^9 \mathrm{N\,m^2/C^2}\)).

Set up equations for initial and final forces

Using the formula for Coulomb's Law and the given information of the initial and final forces, we can set up two equations: Initial force: \[2 \times 10^{-3} = k\frac{q_1q_2}{r^2}\] Final force: \[5 \times 10^{-4} = k\frac{q_1q_2}{(r+0.1)^2}\]

Find the charges on both spheres

First, let's divide the equation for the final force by the equation for the initial force: \[\frac{5 \times 10^{-4}}{2 \times 10^{-3}} = \frac{(r + 0.1)^2}{r^2}\] Upon solving this equation, we find that \(r = 0.2 \mathrm{m}\). Now, plug this value of \(r\) into the equation for the initial force and solve for the product of charges: \[2 \times 10^{-3} = k\frac{q_1q_2}{(0.2)^2}\] \[q_1q_2 = 8.90 \times 10^{-18} \mathrm{C^2}\] Finally, we know that \(q_1 = 2q_2\), therefore \(q_2^2 = 2.97 \times 10^{-18} \mathrm{C^2}\) and \(q_2 = 1.6 \times 10^{-9} \mathrm{C}\). Thus, \(q_1 = 3.2 \times 10^{-9} \mathrm{C}\). The charges on both spheres are: (A) \(1.6 \times 10^{-9} \mathrm{C}\), \(3.2 \times 10^{-9} \mathrm{C}\).

Key concepts

Electrostatics

Electrostatics is the branch of physics that deals with the study of electric charges at rest. This subject examines how charges interact with each other, which can either be through attraction or repulsion. The fundamental principle used to calculate the force between static electric charges is known as Coulomb's Law. The electrostatic force can be compelling, affecting everything from tiny particles to large bodies. Understanding this force is crucial as it plays a role in various physical phenomena and technological applications like capacitors and electrostatic precipitators.

In problems dealing with electrostatics, it's essential to identify the type of charges involved and the nature of the interaction, be it attraction or repulsion, to predict or calculate the forces at work. Electrostatics is pivotal in understanding basics like electric fields, potential energy, and how charged objects can influence each other without any direct contact.

Charge Interaction

Charge interaction is a fundamental concept in electrostatics, where charged objects either attract or repel each other. Like charges repel, whereas opposite charges attract. This principle affects how charges are distributed and reacts in their environment.

The strength of this interaction is expressed through forces calculated using Coulomb's Law. The force between two charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. This means larger charges exert a stronger force, and closer charges interact more strongly than those further apart. Charge interaction is not just theoretical but observable in everyday phenomena like static electricity—consider the zap you feel when touching a metal doorknob after walking across a carpeted room.

Distance and Force Relationship

The relationship between distance and the electrostatic force is key in solving problems involving charges, as evident in the exercise involving the two charged spheres. Coulomb's Law illustrates this with force decreasing as distance increases. Specifically, the force is inversely proportional to the square of the distance. This quadratic relationship means that even small changes in distance can lead to significant changes in force.

This principle helps us understand why charged particles act in certain ways. For instance, two charged particles initially experience a stronger repulsive force when closer. By increasing their separation, as in the exercise example, we significantly reduce the force they exert on each other. Grasping this concept aids in predictions and calculations in physics, making it crucial in both academic settings and practical applications.

Problem-solving in Physics

Problem-solving in physics involves applying theoretical concepts to practical situations, as shown in the textbook exercise provided. This process requires understanding the relationships and formulas that govern physical phenomena. In the case of electrostatics, accurately using Coulomb's Law is critical.

To tackle such problems, begin by identifying all known information and relationships. For example, the charges' values and their proportional relationships, distances involved, and initial conditions are essential data.

• Start by formulating equations based on these conditions, usually derived from fundamental laws like Coulomb's Law.
• Use algebraic manipulation to isolate variables of interest, such as charge values or distances.
• Carefully solve these equations while keeping an eye on units and constants.

By understanding these steps, students can develop a structured approach to tackling physics problems, confident in their ability to analyze and solve complex scenarios.