College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Can you explain how they get to this solution?
![### Page 12 of 274
## Problem 63: Electrostatic Force on a Charge in a Square Configuration
**Problem Statement:**
63. What is the force on the charge \( q \) at the lower-right-hand corner of the square shown here?
**Diagram:**
The diagram shows a square with sides of length \( a \). There are four charges, each denoted as \( q \), positioned at the corners of the square. Specifically:
- The charges are located at the upper-right, upper-left, lower-right, and lower-left corners of the square.
- The sides of the square are labeled as \( a \), and the distance between each charge along the sides is \( a \).
\[
\begin{array}{c|c|c}
& & q \\
a & & a \\
& & q \\
\hline
q & & q \\
\end{array}
\]
**Solution:**
The diagonal is \(\sqrt{2}a\) and the components of the force due to the diagonal charge have a factor \(\cos \theta = \frac{1}{\sqrt{2}}\).
### Detailed Explanation:
1. **Identifying Forces:**
- To find the force on the charge at the lower-right hand corner, consider the contributions from the other three charges.
- Each charge \( q \) exerts a Coulomb force on the charge at the lower-right hand corner.
2. **Calculating Forces:**
- **Side Charges:**
- The force due to the charge directly to the left (at distance \( a \)).
- The force due to the charge directly above (at distance \( a \)).
- **Diagonal Charge:**
- The force due to the charge at the diagonally opposite corner (at distance \( \sqrt{2}a \)).
3. **Components of the Force:**
- For the diagonal charge, resolve the force into its \( x \)-component and \( y \)-component.
- The angle between the side and the diagonal is \( 45^\circ \), leading to components along both axes with a factor of \(\cos 45^\circ = \frac{1}{\sqrt{2}}\).
4. **Vector Addition:**
- Sum the forces vectorially considering the contributions from each charge.
**Summary:**
By calculating the forces from each of the contributing charges and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61f2bd8c-527a-479e-9deb-09c663979694%2F8d934b84-3a0b-4d80-999a-efd4da3105a5%2F2oy0woh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Page 12 of 274
## Problem 63: Electrostatic Force on a Charge in a Square Configuration
**Problem Statement:**
63. What is the force on the charge \( q \) at the lower-right-hand corner of the square shown here?
**Diagram:**
The diagram shows a square with sides of length \( a \). There are four charges, each denoted as \( q \), positioned at the corners of the square. Specifically:
- The charges are located at the upper-right, upper-left, lower-right, and lower-left corners of the square.
- The sides of the square are labeled as \( a \), and the distance between each charge along the sides is \( a \).
\[
\begin{array}{c|c|c}
& & q \\
a & & a \\
& & q \\
\hline
q & & q \\
\end{array}
\]
**Solution:**
The diagonal is \(\sqrt{2}a\) and the components of the force due to the diagonal charge have a factor \(\cos \theta = \frac{1}{\sqrt{2}}\).
### Detailed Explanation:
1. **Identifying Forces:**
- To find the force on the charge at the lower-right hand corner, consider the contributions from the other three charges.
- Each charge \( q \) exerts a Coulomb force on the charge at the lower-right hand corner.
2. **Calculating Forces:**
- **Side Charges:**
- The force due to the charge directly to the left (at distance \( a \)).
- The force due to the charge directly above (at distance \( a \)).
- **Diagonal Charge:**
- The force due to the charge at the diagonally opposite corner (at distance \( \sqrt{2}a \)).
3. **Components of the Force:**
- For the diagonal charge, resolve the force into its \( x \)-component and \( y \)-component.
- The angle between the side and the diagonal is \( 45^\circ \), leading to components along both axes with a factor of \(\cos 45^\circ = \frac{1}{\sqrt{2}}\).
4. **Vector Addition:**
- Sum the forces vectorially considering the contributions from each charge.
**Summary:**
By calculating the forces from each of the contributing charges and
![OpenStax **University Physics Volume II**
**Unit 2: Electricity and Magnetism**
**Chapter 5: Electric Charges and Fields**
\[ \mathbf{F}_{\text{net}} = \left[ k \frac{q^2}{a^2} + k \frac{q^2}{2a^2} \frac{1}{\sqrt{2}} \right] \hat{i} - \left[ k \frac{q^2}{a^2} + k \frac{q^2}{2a^2} \frac{1}{\sqrt{2}} \right] \hat{j} \]
**64. Point charges** \( q_1 = 10 \, \mu C \) **and** \( q_2 = -30 \, \mu C \) **are fixed at** ...](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61f2bd8c-527a-479e-9deb-09c663979694%2F8d934b84-3a0b-4d80-999a-efd4da3105a5%2Fqlhlavp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:OpenStax **University Physics Volume II**
**Unit 2: Electricity and Magnetism**
**Chapter 5: Electric Charges and Fields**
\[ \mathbf{F}_{\text{net}} = \left[ k \frac{q^2}{a^2} + k \frac{q^2}{2a^2} \frac{1}{\sqrt{2}} \right] \hat{i} - \left[ k \frac{q^2}{a^2} + k \frac{q^2}{2a^2} \frac{1}{\sqrt{2}} \right] \hat{j} \]
**64. Point charges** \( q_1 = 10 \, \mu C \) **and** \( q_2 = -30 \, \mu C \) **are fixed at** ...
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