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Chapter 1: Problem 46

Charges on a circular track Suppose three positively charged particles are constrained to move on a fixed circular track. If the charges were all equal, an equilibrium arrangement would obviously be a symmetrical one with the particles spaced \(120^{\circ}\) apart around the circle. Suppose that two of the charges are equal and the equilibrium arrangement is such that these two charges are \(90^{\circ}\) apart rather than \(120^{\circ} .\) What is the relative magnitude of the third charge?

Short Answer

Expert verified
The relative magnitude of the third charge is \(\frac{1}{\sqrt{2}}\) or approximately 0.707 of each of the other two charges.

Step by step solution

01

Understanding the system and drawing the diagram

Imagine a coordinate system with its origin placed in the center of the circle. Place one of the equal charges at the position (1,0) and the other at the position (0,1). The third charge, say \(Q\), is placed at unit distance from the origin along the negative x-axis at the point (-1,0).
02

Applying the electrostatic force equation

The only forces acting on the particles are the electric forces due to the other charges. In equilibrium, the net force on each particle is zero. That means the sum of the forces each charge experiences must be zero. According to Coulomb鈥檚 Law, the force between two charges \(q_1\) and \(q_2\) separated by distance \(r\) is given by \(F = k \cdot \frac{|q_1 \cdot q_2|}{r^2}\), where \(k\) is Coulomb's constant. Here, for simplicity, \(k\) can be set equal to 1 since the question asks only for the relative magnitude of the third charge.
03

Calculating the forces

The first equally charged particle is affected by the forces from \(Q\) (along x-axis) and from the second equally charged particle (along y-axis). As both forces in equilibrium must cancel each other out, the magnitudes of the two forces must be equal: \(\frac{|q \cdot Q|}{1^2} = \frac{|q \cdot q|}{\sqrt{2}^2}\), where \(q\) is the magnitude of each equal charge and \(Q\) is the magnitude of the third charge. This leads to \(|Q| = \frac{q}{\sqrt{2}}\)
04

Interpret the result

The magnitude of \(Q\) is \(\frac{1}{\sqrt{2}}\) times the magnitude of \(q\). Hence, the relative magnitude of the third charge is \(\frac{1}{\sqrt{2}}\) or approximately 0.707 times the magnitude of each of the other two charges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coulomb's Law
Coulomb's Law is a fundamental principle that describes the electrostatic interaction between electrically charged particles. At its core, it states that the electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

This relationship is mathematically represented as \( F = k \cdot \frac{|q_1 \cdot q_2|}{r^2} \), where \( F \) is the magnitude of the electric force, \( q_1 \) and \( q_2 \) are the magnitudes of the electric charges, \( r \) is the distance between the charges, and \( k \) is Coulomb's constant. Understanding this law is critical for solving problems related to the force between charged particles, such as the example provided in the exercise.

Importance in Calculations

The simplicity of Coulomb's Law allows it to be the starting point in calculating electric forces in many physics problems. Once you have identified the charges and their separation, you can determine the magnitude and direction of the electric force exerted by one charge on another. This is particularly useful for students in visualizing how electric charges interact and in predicting the motion of particles under electrostatic forces.
Electric Force
The electric force, an essential concept in electromagnetism, is the force exerted by charged objects on each other. It can be either attractive or repulsive, depending on the nature of the charges involved; like charges repel each other, while opposite charges attract.

In the context of the exercise, we refer to the electric force when discussing how charged particles on a circular track influence each other's motion. When dealing with static situations, or electrostatic equilibrium, the net electric force on each charge is zero. This means that all the individual forces acting on a charge must sum up to zero, leading to a balance.

Equilibrium Considerations

For particles to remain in equilibrium on the track, as in our example, the forces due to the other charges must be counteracted perfectly. In the case where the particles are constrained to a circular track, their arrangement and the symmetry of the system have significant implications on how we determine the final equilibrium state.
Charge Distribution
Charge distribution refers to how electric charge is arranged or spread out in a system. In electrostatics, the charge distribution greatly impacts the electric field and the resultant forces on charges within the field. There are several types of charge distributions, including point charges, linear charges, surface charges, and volume charges, each affecting the electric field in unique ways.

In the exercise, we deal with point charges that are distributed along a circular track. The symmetry or asymmetry in their placement governs the resultant forces and hence affects the equilibrium of the system. When charges are symmetrically placed, they often lead to simpler and more predictable patterns of electric forces and potential.

Effect on Force Calculations

In problems like the one in the exercise, understanding the charge distribution is pivotal for applying the correct form of Coulomb's Law. It enables us to calculate the forces correctly and deduce the nature of the equilibrium that results from the specific arrangement of charges.
Symmetrical Arrangement
Symmetrical arrangement in the context of charge distribution refers to a configuration where charges are spaced evenly around a shape or along an axis. Symmetry simplifies the calculations for the resultant electric force and potential, as it allows one to predict the behavior of the system using principles of balance and equality.

In the exercise example, the initial expectation of an equilibrium arrangement with three equally charged particles would normally be a symmetrical triangle, where each charge is \(120^\circ\) apart. However, the exercise presents an asymmetrical situation where two of the charges are only \(90^\circ\) apart, leading to the need to solve for the relative magnitude of the third charge to maintain equilibrium.

Importance in Problem-Solving

Recognizing symmetrical arrangements is a vital skill as it allows students to apply conservation laws and simplify complex systems for analysis. It is important to note that even in systems that are not perfectly symmetrical, like the one described in the exercise, the principles of equilibrium can still be applied to find the necessary conditions for stability.

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Most popular questions from this chapter

Field in the end face Consider a half-infinite hollow cylindrical shell (that is, one that extends to infinity in one direction) with uniform surface charge density. Show that at all points in the circular end face, the electric field is parallel to the cylinder's axis. Hint: Use superposition, along with what you know about the field from an infinite (in both directions) hollow cylinder.

Potential energy of a cylinder A cylindrical volume of radius \(a\) is filled with charge of uniform density \(\rho\). We want to know the potential energy per unit length of this cylinder of charge, that is, the work done per unit length in assembling it. Calculate this by building up the cylinder layer by layer, making use of the fact that the field outside a cylindrical distribution of charge is the same as if all the charge were located on the axis. You will find that the energy per unit length is infinite if the charges are brought in from infinity, so instead assume that they are initially distributed uniformly over a hollow cylinder with large radius \(R\). Write your answer in terms of the charge per unit length of the cylinder, which is \(\lambda=\rho \pi a^{2}\). (See Exercise \(1.83\) for a different method of solving this problem.)

Field from a semicircle * A thin plastic rod bent into a semicircle of radius \(R\) has a charge \(Q\) distributed uniformly over its length. Find the electric field at the center of the semicircle.

Oscillating in a ring A ring with radius \(R\) has uniform positive charge density \(\lambda\). A particle with positive charge \(q\) and mass \(m\) is initially located at the center of the ring and is then given a tiny kick. If it is constrained to move in the plane of the ring, show that it undergoes simple harmonic motion (for small oscillations), and find the frequency. Hint: Find the potential energy of the particle when it is at a (small) radius, \(r\), by integrating over the ring, and then take the negative derivative to find the force. You will need to use the law of cosines and also the Taylor series \(1 / \sqrt{1+\epsilon} \approx 1-\epsilon / 2+3 \epsilon^{2} / 8\)

Force between two strips ** (a) The two strips of charge shown in Fig. \(1.47\) have width \(b\), infinite height, and negligible thickness (in the direction perpendicular to the page). Their charge densities per unit area are \(\pm \sigma .\) Find the magnitude of the electric field due to one of the strips, a distance \(x\) away from it (in the plane of the page). (b) Show that the force (per unit height) between the two strips equals \(\sigma^{2} b(\ln 2) / \pi \epsilon_{0}\). Note that this result is finite, even though you will find that the field due to a strip diverges as you get close to it
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