Hello, I am having trouble with this problem because I don't know how to do all three of these parts. Can you help me with Part A, PART B, AND PART C and you can label which one is which
Hello, I am having trouble with this problem because I don't know how to do all three of these parts. Can you help me with Part A, PART B, AND PART C and you can label which one is which
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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Hello, I am having trouble with this problem because I don't know how to do all three of these parts. Can you help me with Part A, PART B, AND PART C and you can label which one is which
![**Problem 2:** A hollow cylindrical shell of length \( L \) and radius \( R \) has charge \( Q \) uniformly distributed along its length. What is the electric potential at the center of the cylinder?
a) Compute the surface charge density \( \eta \) of the shell from its total charge and geometrical parameters.
b) Which charge \( dq \) is enclosed in a thin ring of width \( dz \) located at a distance \( z \) from the center of the cylinder (shown in Fig. 2)? Which potential \( dV \) does this ring create at the center? (You need to use the formula derived in the textbook for the potential of a charged ring along its axis).
c) Sum up the contributions from all the rings along the cylinder by integrating \( dV \) with respect to \( z \). Show that
\[
V_{\text{center}} = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{L} \ln \left( \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}} \right)
\]
(The integral that you need to use here is \( \int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2 + a^2}} = \ln(t + \sqrt{t^2 + a^2}) \Big|_{t_1}^{t_2} \)).
---
**Explanation of Figure 2:**
Figure 2 shows a hollow cylindrical shell with length \( L \) and radius \( R \). The shell is oriented along the \( z \)-axis, with the center marked as \( O \). A thin ring element of the shell has a width \( dz \) and is located at a distance \( z \) from the center. The purpose of the ring is to analyze the charge distribution and electric potential at the center due to this infinitesimal element.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c26c01a-85a2-4400-a2e6-3b7f42c87405%2Fad269f88-4018-4b01-94b3-cb07d89e8862%2Fsgudesr_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 2:** A hollow cylindrical shell of length \( L \) and radius \( R \) has charge \( Q \) uniformly distributed along its length. What is the electric potential at the center of the cylinder?
a) Compute the surface charge density \( \eta \) of the shell from its total charge and geometrical parameters.
b) Which charge \( dq \) is enclosed in a thin ring of width \( dz \) located at a distance \( z \) from the center of the cylinder (shown in Fig. 2)? Which potential \( dV \) does this ring create at the center? (You need to use the formula derived in the textbook for the potential of a charged ring along its axis).
c) Sum up the contributions from all the rings along the cylinder by integrating \( dV \) with respect to \( z \). Show that
\[
V_{\text{center}} = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{L} \ln \left( \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}} \right)
\]
(The integral that you need to use here is \( \int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2 + a^2}} = \ln(t + \sqrt{t^2 + a^2}) \Big|_{t_1}^{t_2} \)).
---
**Explanation of Figure 2:**
Figure 2 shows a hollow cylindrical shell with length \( L \) and radius \( R \). The shell is oriented along the \( z \)-axis, with the center marked as \( O \). A thin ring element of the shell has a width \( dz \) and is located at a distance \( z \) from the center. The purpose of the ring is to analyze the charge distribution and electric potential at the center due to this infinitesimal element.
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