/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Four identical charges \(Q\) are... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 21: Problem 23

Four identical charges \(Q\) are placed at the corners of a square of side \(L .\) (a) In a free-body diagram, show all of the forces that act on one of the charges. (b) Find the magnitude and direction of the total force exerted on one charge by the other three charges.

Short Answer

Expert verified
The magnitude of the force on one charge is \( F = \sqrt{2} \left(\frac{k \, Q^2}{L^2} + \frac{k \, Q^2}{2\sqrt{2}L^2}\right) \) and it points at 45掳 to the x-axis.

Step by step solution

01

Understanding the Forces on One Charge

First, identify the position of the charges. Let's say you choose the charge located at the lower-left corner of the square. The other three charges are positioned at the lower-right, upper-left, and upper-right corners. Each one exerts an electrostatic force on the selected charge.
02

Drawing the Forces on the Charge

In the free-body diagram for the charge at the lower-left corner, draw:1. A force \( F_1 \) due to the charge at the lower-right corner, acting to the right.2. A force \( F_2 \) due to the charge at the upper-left corner, acting upwards.3. A force \( F_3 \) exerted by the charge at the upper-right corner, directed along the diagonal towards the charge's position.
03

Calculating Individual Forces

Use Coulomb's Law, \( F = \frac{{k \, |Q|^2}}{{r^2}} \), to compute each force:- For \( F_1 \): \( r = L \). Thus, \( F_1 = \frac{{k \, Q^2}}{{L^2}} \).- For \( F_2 \): \( r = L \). Thus, \( F_2 = \frac{{k \, Q^2}}{{L^2}} \).- For \( F_3 \): \( r = \sqrt{2}L \) (diagonal of square). Thus, \( F_3 = \frac{{k \, Q^2}}{{2L^2}} \).
04

Finding Components of Diagonal Force

Break the diagonal force \( F_3 \) into components. Both the x and y components are: \( F_{3x} = F_{3y} = \frac{{k \, Q^2}}{{2L^2 \sqrt{2}}} = \frac{{k \, Q^2}}{{2 \sqrt{2} L^2}} \).
05

Calculating the Total Force in x-Direction

Sum the x-component forces:- Total x-component force \( F_{x,total} = F_1 + F_{3x} = \frac{{k \, Q^2}}{{L^2}} + \frac{{k \, Q^2}}{{2\sqrt{2}L^2}} \).
06

Calculating the Total Force in y-Direction

Sum the y-component forces:- Total y-component force \( F_{y,total} = F_2 + F_{3y} = \frac{{k \, Q^2}}{{L^2}} + \frac{{k \, Q^2}}{{2\sqrt{2}L^2}} \).
07

Finding the Magnitude of the Total Force

Use Pythagoras theorem to compute the total force magnitude:- \[ F_{total} = \sqrt{(F_{x,total})^2 + (F_{y,total})^2} = \sqrt{\left(\frac{{k \, Q^2}}{{L^2}} + \frac{{k \, Q^2}}{{2\sqrt{2}L^2}}\right)^2 + \left(\frac{{k \, Q^2}}{{L^2}} + \frac{{k \, Q^2}}{{2\sqrt{2}L^2}}\right)^2} \].
08

Direction of the Total Force

The direction of the total force is along the line at 45 degrees to the x-axis (due to symmetry in contribution of forces in x and y directions).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Electrostatic Force
Electrostatic force is the force of attraction or repulsion that acts between charged particles. It is a fundamental concept in physics and is described by Coulomb's Law. Coulomb's Law states that the electrostatic force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, it can be written as \[ F = \frac{{k \, |Q_1 Q_2|}}{{r^2}} \] where:
  • \( F \) is the magnitude of the electrostatic force,
  • \( k \) is the Coulomb's constant \( (8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2) \),
  • \( Q_1 \) and \( Q_2 \) are the magnitudes of the charges,
  • \( r \) is the distance between the charges.
For charges of the same sign, the force is repulsive, causing them to move apart. For charges of opposite signs, the force is attractive, making them move toward one another. This principle can be seen in situations where multiple charges interact, such as charges located at the corners of a square. The interaction of each charge with every other charge must be considered to determine the net force acting on any particular charge.
Free-Body Diagram
A free-body diagram is a graphical representation used to visualize the forces acting on an object. In physics, it is a valuable tool to analyze how the forces interact and influence the motion or static equilibrium of an object. In strength and direction, each force is drawn as an arrow pointing in the direction the force acts. In the context of electrostatics, a free-body diagram helps to identify and analyze the forces on a charged particle due to other surrounding charges. For example, when considering one of four identical charges placed at the corners of a square, it is crucial to understand how it interacts with each of the other three charges. - Direction and Magnitude: Draw arrows for each force based on their direction and magnitude, calculated using Coulomb's law. - Superposition Principle: The net force acting on the charge is the vector sum of all the forces from the surrounding charges. Creating a free-body diagram requires understanding of geometry and trigonometry to accurately depict forces, especially for non-linear arrangements such as diagonal forces.
Vector Components
Vector components are essential tools in analyzing forces acting at angles, as they allow us to break down a vector into its horizontal (x) and vertical (y) components. This simplification makes it easier to perform calculations and understand the net effect of multiple forces on an object. For example, in a square configuration of charges, any force acting diagonally needs to be broken into its x and y components to better understand the total force acting along conventional axes. - X and Y Components: To find the components of a diagonal vector, you often use trigonometric ratios, such as sine and cosine, or symmetry arguments. For a diagonal force \( F_3 \) along a line at an angle, its components can be calculated as: \[ F_{3x} = F_3 \cos(45^\circ) = \frac{{F_3}}{{\sqrt{2}}} \] \[ F_{3y} = F_3 \sin(45^\circ) = \frac{{F_3}}{{\sqrt{2}}} \]- Purpose: By computing vector components, you can sum the forces along each axis separately, greatly simplifying the problem.This approach is integral to finding the resultant force and direction in complex systems involving electrostatic forces and other vector quantities.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A proton is projected into a uniform electric field that points vertically upward and has magnitude \(E\) . The initial velocity of the proton has a magnitude \(v_{0}\) and is directed at an angle \(\alpha\) below the horizontal. (a) Find the maximum distance \(h_{\text { max }}\) that the proton descends vertically below its initial elevation. You can ignore gravitational forces. ( \(b\) ) After what horizontal distance \(d\) does the proton return to its original elevation? (c) Sketch the trajectory of the proton. (d) Find the numerical values of \(h_{\text { max }}\) and \(d\) if \(E=500 \mathrm{N} / \mathrm{C}, v_{0}=4.00 \times 10^{5} \mathrm{m} / \mathrm{s},\) and \(\alpha=30.0^{\circ} .\)

Two small aluminum spheres, each having mass 0.0250 \(\mathrm{kg}\) are separated by \(80.0 \mathrm{cm} .\) (a) How many electrons does each sphere contain? (The atomic mass of aluminum is 26.982 \(\mathrm{g} / \mathrm{mol}\) , and its atomic number is \(13 . )\) (b) How many electrons would have to be removed from one sphere and added to the other to cause an attractive force between the spheres of magnitude \(1.00 \times 10^{4} \mathrm{N}\) (roughly 1 ton \() ?\) Assume that the spheres may be treated as point charges. (c) What fraction of all the electrons in each sphere does this represent?

Two small spheres spaced 20.0 \(\mathrm{cm}\) apart have equal charge. How many excess electrons must be present on each sphere if the magnitude of the force of repulsion between them is \(4.57 \times 10^{-21} \mathrm{N} ?\)

Lightning occurs when there is a flow of electric charge (principally electrons) between the ground and a thundercloud. The maximum rate of charge flow in a lightning bolt is about \(20,000 \mathrm{C} / \mathrm{s} ;\) this lasts for 100\(\mu \mathrm{s}\) or less. How much charge flows between the ground and the cloud in this time? How many electrons flow during this time?

A positive point charge \(q\) is placed on the \(+y\) -axis at \(y=a\) and a negative point clarge \(-q\) is placed on the \(-y\) -axis at \(y=-a\) . A negative point charge \(-Q\) is located at some point on the \(+x\) -axis, (a) In a free- body diagram, show the forces that act on the charge \(-Q\) . (b) Find the \(x\) - and \(y\) -components of the net force that the two charges \(q\) and \(-q\) exert on \(-Q .\) (Your answer should involve only \(k, q, Q, a\) and the coordinate \(x\) of the third charge. \()\) (c) What is the net force on the charge \(-Q\) when it is at the origin \((x=0)\) ? (d) Graph the \(y\) -component of the net force on the charge \(-Q\) as a function of \(x\) for values of \(x\) between \(-4 a\) and \(+4 a\) .
See all solutions

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Read Explanation

Rotational Dynamics

Read Explanation

Linear Momentum

Read Explanation

Electromagnetism

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.