7. A rod of line t has a total charge of Q and a linear charge density of A, find the electric potential at point P due to the rod dq dx= The electric potential dV at P due to dq at x is: dV = k- dq Where dq = Adx and r Integrate over the charge distribution on the line: (Show your work below) bp v= [ = Adx k- %3D va' + x kQ 1+ Va? +1? - In 1 (1+ va² +1? = kAln a a
7. A rod of line t has a total charge of Q and a linear charge density of A, find the electric potential at point P due to the rod dq dx= The electric potential dV at P due to dq at x is: dV = k- dq Where dq = Adx and r Integrate over the charge distribution on the line: (Show your work below) bp v= [ = Adx k- %3D va' + x kQ 1+ Va? +1? - In 1 (1+ va² +1? = kAln a a
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)...
Related questions
Question
Help with this question please.
![### Problem Statement:
7. **Given:**
- A rod of length \( l \) has a total charge \( Q \) and a linear charge density of \( \lambda \).
- **Objective:** Find the electric potential at point \( P \) due to the rod.
### Diagram Description:
The diagram illustrates a rod aligned along the x-axis, with a point \( P \) located on the y-axis at a distance \( a \) from the origin \( O \). The rod extends from \( O \) to length \( l \).
### Explanation:
An infinitesimal charge element \( dq \) is considered at a distance \( x \) along the rod. The distance \( r \) from this infinitesimal element to point \( P \) is given by:
\[ r = \sqrt{a^2 + x^2} \]
### Electric Potential Calculation:
The electric potential \( dV \) at \( P \) due to \( dq \) at \( x \) is:
\[ dV = k \frac{dq}{r} \]
Where:
\[ dq = \lambda dx \quad \text{and} \quad r = \sqrt{a^2 + x^2} \]
### Integration Over the Charge Distribution:
To find the total electric potential \( V \) at point \( P \), we integrate along the length of the rod from \( x = 0 \) to \( x = l \):
\[ V = \int_0^l k \frac{dq}{r} = \int_0^l k \frac{\lambda dx}{\sqrt{a^2 + x^2}} \]
### Solution:
Combining the terms and solving the integral:
\[
V = k \lambda \int_0^l \frac{dx}{\sqrt{a^2 + x^2}}
\]
This integral evaluates to:
\[
V = k \lambda \left. \ln \left( \frac{x + \sqrt{a^2 + x^2}}{a} \right) \right|_0^l
\]
After applying the limits:
\[
V = k \lambda \left[ \ln \left( \frac{l + \sqrt{a^2 + l^2}}{a} \right) - \ln \left( \frac{0 + \sqrt{a^2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe167cdfc-6687-4357-a9c7-ca435187bf42%2F5a995e6e-64e4-40e6-9406-9879f956eb60%2Fqvht3px_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
7. **Given:**
- A rod of length \( l \) has a total charge \( Q \) and a linear charge density of \( \lambda \).
- **Objective:** Find the electric potential at point \( P \) due to the rod.
### Diagram Description:
The diagram illustrates a rod aligned along the x-axis, with a point \( P \) located on the y-axis at a distance \( a \) from the origin \( O \). The rod extends from \( O \) to length \( l \).
### Explanation:
An infinitesimal charge element \( dq \) is considered at a distance \( x \) along the rod. The distance \( r \) from this infinitesimal element to point \( P \) is given by:
\[ r = \sqrt{a^2 + x^2} \]
### Electric Potential Calculation:
The electric potential \( dV \) at \( P \) due to \( dq \) at \( x \) is:
\[ dV = k \frac{dq}{r} \]
Where:
\[ dq = \lambda dx \quad \text{and} \quad r = \sqrt{a^2 + x^2} \]
### Integration Over the Charge Distribution:
To find the total electric potential \( V \) at point \( P \), we integrate along the length of the rod from \( x = 0 \) to \( x = l \):
\[ V = \int_0^l k \frac{dq}{r} = \int_0^l k \frac{\lambda dx}{\sqrt{a^2 + x^2}} \]
### Solution:
Combining the terms and solving the integral:
\[
V = k \lambda \int_0^l \frac{dx}{\sqrt{a^2 + x^2}}
\]
This integral evaluates to:
\[
V = k \lambda \left. \ln \left( \frac{x + \sqrt{a^2 + x^2}}{a} \right) \right|_0^l
\]
After applying the limits:
\[
V = k \lambda \left[ \ln \left( \frac{l + \sqrt{a^2 + l^2}}{a} \right) - \ln \left( \frac{0 + \sqrt{a^2
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 4 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Recommended textbooks for you
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:
9781337553278
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:
9780321820464
Author:
Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:
Addison-Wesley
College Physics: A Strategic Approach (4th Editio…
Physics
ISBN:
9780134609034
Author:
Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:
PEARSON