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Chapter 18: Problem 68

Two objects are identical and small enough that their sizes can be ignored relative to the distance between them, which is \(0.200 \mathrm{~m}\). In a vacuum, each object carries a different charge, and they attract each other with a force of \(1.20 \mathrm{~N}\). The objects are brought into contact, so the net charge is shared equally, and then they are returned to their initial positions. Now it is found that the objects repel one another with a force whose magnitude is equal to that of the initial attractive force. What is the initial charge on each object? Note that there are two answers.

Short Answer

Expert verified
Initial charges are approximately \( q_1 = 1.096 \times 10^{-5} \, \text{C} \) and \( q_2 = 3.66 \times 10^{-6} \, \text{C} \).

Step by step solution

01

Apply Coulomb's Law for Initial Charge Conditions

Initially, the force between the two objects is attractive and has a magnitude of 1.20 N. Using Coulomb's Law, \( F = \frac{k |q_1 q_2|}{r^2} \), where \( k \) is the Coulomb's constant \( 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2 \), the distance \( r = 0.200 \, \text{m} \), and \( F = 1.20 \, \text{N} \), we can express the relationship between the charges as:\[ |q_1 q_2| = \frac{F r^2}{k} = \frac{1.20 \times (0.200)^2}{8.99 \times 10^9} \approx 5.34 \times 10^{-12} \, \text{C}^2 \]
02

Calculate Combined Charge After Contact and Redistribution

When the objects are brought into contact, their charges are combined and split equally, leading to equal final charges \( q \) on each object. The combined charge is \( q_1 + q_2 \), and each object gets \( \frac{q_1 + q_2}{2} = q \). Thus, the net charge becomes \( q_1 + q_2 = 2q \).
03

Use New Force Condition for Repelling Charges

After redistribution, the objects repel each other with the same magnitude of force, 1.20 N. Using Coulomb's Law for this repulsion, \( F = \frac{k q^2}{r^2} = 1.20 \). Solving for the net charge \( q \), we have:\[ q^2 = \frac{1.20 \times (0.200)^2}{8.99 \times 10^9} \approx 5.34 \times 10^{-12} \, \text{C}^2 \]The shared charge \( q = \pm \sqrt{5.34 \times 10^{-12}} \approx \pm 7.31 \times 10^{-6} \, \text{C} \).
04

Determine Initial Charges Using System of Equations

Given \( q_1 q_2 = 5.34 \times 10^{-12} \) and \( q_1 + q_2 = 2q = 2 \times 7.31 \times 10^{-6} = 1.462 \times 10^{-5} \), use the system of equations:1. \( q_1 q_2 = 5.34 \times 10^{-12} \)2. \( q_1 + q_2 = 1.462 \times 10^{-5} \)These can be solved using the quadratic equation \( x^2 - (q_1 + q_2)x + q_1 q_2 = 0 \), where \( x = q_1 \) or \( q_2 \). The solutions are:\[ q_1, q_2 = \frac{1.462 \times 10^{-5} \pm \sqrt{(1.462 \times 10^{-5})^2 - 4 \times 5.34 \times 10^{-12}}}{2} \]Calculating the values:\[ q_1, q_2 = \frac{1.462 \times 10^{-5} \pm 7.299 \times 10^{-6}}{2} \]This results in\[ q_1 = 1.096 \times 10^{-5} \, \text{C}, \ q_2 = 3.66 \times 10^{-6} \, \text{C} \] or vice versa.

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Key Concepts

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

Electric Charge Distribution
Electric charge distribution refers to how electric charges are arranged in a system. It is crucial in determining how charged objects interact. When two charged objects are brought together and then separated, the charges might be redistributed between them. In the original exercise, this concept is demonstrated when the objects are initially charged, brought into contact to share their charge equally, and then returned to their original positions.

Upon contact, the charges on both objects combine to form a total charge, which is then shared equally. This means that regardless of the initial distribution, the final charge on each object becomes half of this combined charge. This sharing is based on the principle of charge conservation, which states that the total charge in an isolated system remains constant.

Understanding electric charge distribution is essential in solving problems related to forces between charged objects. Charge distribution doesn't just affect electrical forces; it can also impact electrical fields and potential energies within a given space.
Force Between Charged Objects
The force between charged objects is calculated using Coulomb's Law. This law dictates that the force (\( F \)) between two point charges (\( q_1 \) and \( q_2 \)) separated by a distance (\( r \)) is given by the formula: \[ F = \frac{k |q_1 q_2|}{r^2} \]where \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2 \).

In our specific exercise, initially, the force is attractive, meaning the charges are opposite (one positive, one negative). Upon contact and redistribution, the charges become the same鈥攅ither both positive or both negative鈥攔esulting in a repulsive force. Either way, the force magnitude remains the same due to the charge balance and distance, demonstrating the symmetry in charge interactions as governed by Coulomb鈥檚 Law.

Key points explained by this include:
  • Forces between charges depend on both the magnitude of charges and their separation distance.
  • Like charges repel, unlike charges attract.
  • The direction of the force aligns along the line connecting the charged objects.
Quadratic Equations in Physics
Quadratic equations often appear in physics when dealing with problems that involve systems of equations with unknowns, such as charge values or projectile motions. In the given exercise, the initial and redistributed charges form a system that can be expressed as quadratic equations.

The problem gives two equations involving charges \( q_1 \) and \( q_2 \). The equations to be solved are:
  • \( q_1 q_2 = 5.34 \times 10^{-12} \) 鈥� represents the product of the charges initially
  • \( q_1 + q_2 = 1.462 \times 10^{-5} \) 鈥� represents the sum of charges after redistribution
These equations lead to a quadratic equation in the form:\[ x^2 - (q_1 + q_2)x + q_1 q_2 = 0 \]Quadratic equations are solved using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In the context of charge problems, solving the quadratic equation helps in determining the precise initial values of the charges \( q_1 \) and \( q_2 \). Quadratic equations are a powerful tool in physics, providing solutions to complex relationships and interactions among variables.

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Most popular questions from this chapter

A vertical wall \((5.9 \mathrm{~m} \times 2.5 \mathrm{~m})\) in a house faces due east. A uniform electric field has a magnitude of \(150 \mathrm{~N} / \mathrm{C}\). This field is parallel to the ground and points \(35^{\circ}\) north of east. What is the electric flux through the wall?

Suppose you want to neutralize the V gravitational attraction between the earth and the moon by placing equal amounts of charge on each. (a) Should the charges be both positive, both negative, or one positive and the other negative? Why? (b) Do you need to know the distance between the earth and the moon to find the magnitude of the charge? Why or why not?

Consider three identical metal spheres, \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). Sphere A carries a charge of \(+5 q .\) Sphere \(\mathrm{B}\) carries a charge of \(-q\). Sphere \(\mathrm{C}\) carries no net charge. Spheres \(\mathrm{A}\) and \(\mathrm{B}\) are touched together and then separated. Sphere \(\mathrm{C}\) is then touched to sphere \(A\) and separated from it. Last, sphere \(C\) is touched to sphere \(B\) and separated from it. (a) How much charge ends up on sphere \(\mathrm{C}\) ? What is the total charge on the three spheres (b) before they are allowed to touch each other and (c) after they have touched?

Consult Concept Simulation 18.1 at for insight into this problem. Three charges are fixed to an \(x, y\) coordinate system. A charge of \(+18 \mu \mathrm{C}\) is on the \(y\) axis at \(y=+3.0 \mathrm{~m}\). A charge of \(-12 \mu \mathrm{C}\) is at the origin. Last, a charge of \(+45 \mu \mathrm{C}\) is on the \(x\) axis at \(x=+3.0 \mathrm{~m}\). Determine the magnitude and direction of the net electrostatic force on the charge at \(x=+3.0 \mathrm{~m}\). Specify the direction relative to the \(-x\) axis.

Four identical metallic objects carry the following charges: \(+1.6,+6.2,-4.8,\) and \(-9.4 \mu \mathrm{C} .\) The objects are brought simultaneously into contact, so that each touches the others. Then they are separated, (a) What is the final charge on each object? (b) How many electrons (or protons) make up the final charge on each object?
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