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Chapter 21: Problem 64

Two small, positively charged spheres have a combined charge of \(5.0 \times 10^{-5} \mathrm{C}\). If each sphere is repelled from the other by an electrostatic force of \(1.0 \mathrm{~N}\) when the spheres are \(2.0 \mathrm{~m}\) apart, what is the charge on the sphere with the smaller charge?

Short Answer

Expert verified
The sphere with the smaller charge has approximately \(1.67 \times 10^{-5} \, \text{C}\).

Step by step solution

01

Understand Coulomb's Law

Coulomb's law describes the electrostatic force between two charged objects as \( F = k \frac{|q_1 q_2|}{r^2} \), where \( F \) is the force, \( k \) is Coulomb's constant \( (8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2) \), \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between the charges.
02

Set Up the Equation to Solve for Charge

We are given that the force \( F = 1.0 \, \text{N} \), the distance \( r = 2.0 \, \text{m} \), and combined charge \( q_1 + q_2 = 5.0 \times 10^{-5} \, \text{C} \). Using Coulomb's law: \[ 1.0 = 8.99 \times 10^9 \times \frac{q_1 q_2}{4} \]. We need to express one charge in terms of the other.
03

Express Charges in Terms of Each Other

Let \( q_1 = x \) and \( q_2 = 5.0 \times 10^{-5} - x \). Substitute these into the Coulomb's law equation: \[ 1.0 = 8.99 \times 10^9 \times \frac{x(5.0 \times 10^{-5} - x)}{4} \].
04

Solve the Quadratic Equation

Simplify the equation to a quadratic form: \[ 4 = 8.99 \times 10^9 (5.0 \times 10^{-5}x - x^2) \]. Solve for \( x \) using the quadratic formula \( ax^2 + bx + c = 0 \). Calculate the discriminant and solve for \( x \).
05

Determine the Smaller Charge

Calculate the two possible values for \( x \). The smaller of the two will correspond to the sphere with the smaller charge. Substitute back to check consistency with the total charge constraint.

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Key Concepts

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

Electrostatic Force
Every charged object exerts an electrostatic force on another charged object. This force can be either attractive or repulsive depending on the types of charges involved. Like charges repel each other while opposite charges attract. The strength of this force is calculated using Coulomb's Law, which is expressed by the formula:
  • \( F = k \frac{|q_1 q_2|}{r^2} \)
Here, \( F \) is the electrostatic force in newtons, \( k \) is the Coulomb's constant with a value of \( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \), \( q_1 \) and \( q_2 \) are the magnitudes of the charges in coulombs, and \( r \) is the distance between the charges in meters.
This law allows us to figure out how much force is acting between two charges based on how much charge there is and how far apart they are. It's a key concept to understand any interaction between charged particles.
Quadratic Equation
A quadratic equation is a type of polynomial equation that is pivotal in solving many physics and mathematical problems, including those involving electrostatic forces when charges are expressed in terms of each other. The standard form of a quadratic equation is
  • \( ax^2 + bx + c = 0 \)
In our exercise, we came across a situation where we had to find the value of a charge, \( x \), leading us to a quadratic equation after substituting the given values into Coulomb's Law.
Solving a quadratic equation often requires using the quadratic formula:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a \), \( b \), and \( c \) are coefficients from the equation.
Understanding quadratic equations is crucial because they frequently appear in various problems, from projectile motions in physics to calculating optimal sales in economics.
Charge Distribution
Charge distribution refers to how electric charge is distributed among objects. When dealing with problems involving multiple charges, it's essential to understand how these charges affect each other and how they are distributed. In our example, the total charge of two spheres was given, which is a vital piece of information needed for solving the problem.
By recognizing the combined charge of two objects, we can set up equations that express the individual charges in terms of one another. In simple terms, if you know the total charge, you can easily distribute that charge among the objects based on additional given conditions, such as the force outcome of their interaction.
  • Overall Charge: Sum of all individual charges.
  • Charge Balance: Ensures that when distributed among multiple objects, the total charge accounts for all individual charges as a sum.
Understanding how to manage and solve for charge distributions is essential in electrical engineering and physics, ensuring that systems operate under the correct parameters.

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Most popular questions from this chapter

Two tiny, spherical water drops, with identical charges of \(-1.00 \times 10^{-16} \mathrm{C}\), have a center-to-center separation of \(1.00 \mathrm{~cm}\). (a) What is the magnitude of the electrostatic force acting between them? (b) How many excess electrons are on each drop, giving it its charge imbalance?

Four particles form a square. The charges are \(q_{1}=\) \(q_{4}=Q\) and \(q_{2}=q_{3}=q .\) (a) What is \(Q / q\) if the net electrostatic force on particles 1 and 4 is zero? (b) Is there any value of \(q\) that makes the net electrostatic force on each of the four particles zero? Explain.

A nonconducting spherical shell, with an inner radius of \(4.0\) \(\mathrm{cm}\) and an outer radius of \(6.0 \mathrm{~cm}\), has charge spread nonuniformly through its volume between its inner and outer surfaces. The volume charge density \(\rho\) is the charge per unit volume, with the unit coulomb per cubic meter. For this shell \(\rho=b / r\), where \(r\) is the distance in meters from the center of the shell and \(b=3.0 \mu \mathrm{C} / \mathrm{m}^{2} .\) What is the net charge in the shell?

A particle of charge \(+3.00 \times 10^{-6} \mathrm{C}\) is \(12.0 \mathrm{~cm}\) distant from a second particle of charge \(-1.50 \times 10^{-6} \mathrm{C} .\) Calculate the magnitude of the electrostatic force between the particles.

A current of \(0.300\) A through your chest can send your heart into fibrillation, ruining the normal rhythm of heartbeat and disrupting the flow of blood (and thus oxygen) to your brain. If that current persists for \(2.00 \mathrm{~min}\), how many conduction electrons pass through your chest?
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