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?
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?
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 only need help with part A and PART B can you label which one is which thank you.
![**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
\[
V_{\text{center}} = \frac{1}{4 \pi \epsilon_0} \frac{Q}{L} \ln \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}}
\]
(The integral that you need to use here is \( \int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2+a^2}} = \ln(t + \sqrt{t^2 + a^2}) \bigg|_{t_1}^{t_2} \))
**Diagram (Fig. 2 Explanation)**:
- The cylindrical shell is depicted with length \( L \) and radius \( R \).
- A thin ring of width \( dz \) is shown at a distance \( z \) from the center of the cylinder.
- The axis of the cylinder is labeled, with the origin \( O \) at the center of the cylinder.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2877aac-ba93-4be6-806c-de0e3196a7e6%2F2bfaf5b9-e41b-4b8b-9165-f0fb03854419%2Fc6v3z6g_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
\[
V_{\text{center}} = \frac{1}{4 \pi \epsilon_0} \frac{Q}{L} \ln \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}}
\]
(The integral that you need to use here is \( \int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2+a^2}} = \ln(t + \sqrt{t^2 + a^2}) \bigg|_{t_1}^{t_2} \))
**Diagram (Fig. 2 Explanation)**:
- The cylindrical shell is depicted with length \( L \) and radius \( R \).
- A thin ring of width \( dz \) is shown at a distance \( z \) from the center of the cylinder.
- The axis of the cylinder is labeled, with the origin \( O \) at the center of the cylinder.
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