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Chapter 18: Problem 26

There are four charges, each with a magnitude of 2.0 C. Two are positive and two are negative. The charges are fixed to the corners of a 0.30-m square, one to a corner, in such a way that the net force on any charge is directed toward the center of the square. Find the magnitude of the net electrostatic force experienced by any charge.

Short Answer

Expert verified
The net electrostatic force on each charge is zero due to symmetry.

Step by step solution

01

Understand the Problem Setup

We have four charges at the corners of a square. Each charge is equal in magnitude (2.0 C), but two are positive and two are negative. The charges are arranged such that the net force on any charge points toward the square's center.
02

Determine Charges' Arrangement

In order to ensure that the net force on each charge points towards the center, arrange the charges alternately. That means if a positive charge is at one corner, the adjacent corners should have negative charges, and the opposite corner should have a positive charge.
03

Calculate Pairwise Force

The electrostatic force between any two charges is given by Coulomb's law:\[ F = \frac{k |q_1 q_2|}{r^2} \]where \( k \) is Coulomb's constant \( 8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2 \), \( |q_1 q_2| \) is the product of the magnitudes of two charges, and \( r \) is the distance between them. Here \( r = 0.30 \, \text{m} \).
04

Calculate the Resultant of Diagonal Forces

Calculate the force between the two charges diagonally opposite to each other:\[ F_{\text{diag}} = \frac{k q^2}{(\sqrt{2} \cdot 0.30)^2} = \frac{8.99 \times 10^9 \, (2.0)^2}{(0.30 \sqrt{2})^2} \]
05

Calculate Force Contribution from Adjacent Charges

Each charge experiences two forces from adjacent charges at 90 degrees to each other in the square configuration. Calculate these using Coulomb's law, and note their perpendicular arrangement results in a resultant force that points inward.
06

Vector Addition of Forces

The net electrostatic force on each charge is determined by vector addition of the three forces (one from each pair). The force from opposite diagonal charge, and two forces from adjacent charges. The diagonal contribution is bigger and generally directs towards the center.
07

Compute Net Force Magnitude

The net force can be calculated by considering contributions from all charge interactions. Given symmetry, compute the effective force considering charge interactions using superimposition principles.

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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 in electrostatics that quantifies the electric force between two point charges. It tells us how strong the force is and the direction in which it acts. The force between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

This can be expressed mathematically as:
  • \[ F = \frac{k |q_1 q_2|}{r^2} \]
Where:
  • \( F \) is the magnitude of the force between the charges.
  • \( k \) is Coulomb's constant \( 8.99 \times 10^9 \, \text{Nm}^2 / \text{C}^2 \).
  • \( |q_1 q_2| \) is the product of the magnitudes of the two charges.
  • \( r \) is the distance between the charges.
Coulomb's Law helps determine how charges interact and is crucial for solving problems involving multiple charges like in this square configuration.
Electric Charge Interactions
Electric charge interactions are all about how charges affect one another. Opposite charges (positive and negative) attract, while like charges (positive-positive or negative-negative) repel each other. This interaction creates a force that can cause objects to move or influence their surroundings.

In the given problem, we have two positive and two negative charges. They are arranged alternately at the corners of a square. This specific arrangement ensures that the forces on each charge point towards the center of the square.
  • Positive charges will attract negative charges diagonally opposite them.
  • Each charge also interacts with its neighboring charges, resulting in forces that create a net inward direction.
Understanding these interactions helps in predicting how the charges will move or exert force on one another.
Vector Addition of Forces
Vector addition of forces is a method used to calculate the net force when multiple forces act on an object. Since forces are vectors, they have both a magnitude and a direction. In this problem, each charge is influenced by forces from three other charges:

  • Two diagonal forces acting between opposite charges.
  • Two adjacent forces from neighboring charges.
To find the net force on any charge, we need to add these vectorially. Typically, this involves considering both the magnitude and direction of each force.
  • Forces from diagonal charges are usually stronger because they are not canceled by perpendicular components.
  • Forces from adjacent charges combine perpendicularly, resulting in a net force pointing towards the center of the square.
By using vector addition, you can determine the overall force experienced by each charge, crucial for predicting the behavior of the configuration.

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Most popular questions from this chapter

Two spherical shells have a common center. A \(-1.6 \times 10^{-6} \mathrm{C}\) charge is spread uniformly over the inner shell, which has a radius of \(0.050 \mathrm{m} . \mathrm{A}+5.1 \times 10^{-6} \mathrm{C}\) charge is spread uniformly over the outer shell, which has a radius of 0.15 \(\mathrm{m}\) . Find the magnitude and direction of the electric field at a distance (measured from the common center) of (a) \(0.20 \mathrm{m},\) (b) \(0.10 \mathrm{m},\) and \((\mathrm{c}) 0.025 \mathrm{m} .\)

A charge \(Q\) is located inside a rectangular box. The electric flux through each of the six surfaces of the box is: \(\Phi_{1}=+1500 \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}\) \(\Phi_{2}=+2200 \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}, \Phi_{3}=+4600 \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}, \Phi_{4}=-1800 \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}\) \(\Phi_{5}=-3500 \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C},\) and \(\Phi_{6}=-5400 \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C} .\) What is \(Q ?\)

A spherical surface completely surrounds a collection of charges.Find the electric flux through the surface if the collection consists of (a) a single \(+3.5 \times 10^{-6} \mathrm{C}\) charge, (b) a single \(-2.3 \times 10^{-6} \mathrm{C}\) charge, and \((\mathrm{c})\) both of the charges in (a) and (b).

ssm mmh Two tiny spheres have the same mass and carry charges of the same magnitude. The mass of each sphere is \(2.0 \times 10^{-6} \mathrm{kg}\) . The gravitational force that each sphere exerts on the other is balanced by the electric force. (a) What alyebraic signs can the charges have? (b) Determine the charge magnitude.

A plate carries a charge of \(-3.0 \mu \mathrm{C},\) while a rod carries a charge of \(+2.0 \mu \mathrm{C}\) . How many electrons must be transferred from the plate to the rod, so that both objects have the same charge?
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