**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: \[ V_{\text{center}} = \frac{1}{4\pi \varepsilon_0 L} Q \ln \left( \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}} \right). \] (The integral that you need to use here is \( \int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2+a^2}} = \ln(t + \sqrt{t^2+a^2}) \big|_{t_1}^{t_2} \).) **Figure 2: The scheme for Problem 2** The figure illustrates a hollow cylindrical shell. The cylinder has a radius \( R \) and a length \( L \). It shows a small section at a distance \( z \) from the center of the cylinder, labeled \( dz \). The center of the cylinder is marked \( O \). The figure helps in visualizing how a thin ring of width \( dz \) is considered at a distance \( z \) for calculating the electric potential at the center due to this ring.
**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: \[ V_{\text{center}} = \frac{1}{4\pi \varepsilon_0 L} Q \ln \left( \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}} \right). \] (The integral that you need to use here is \( \int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2+a^2}} = \ln(t + \sqrt{t^2+a^2}) \big|_{t_1}^{t_2} \).) **Figure 2: The scheme for Problem 2** The figure illustrates a hollow cylindrical shell. The cylinder has a radius \( R \) and a length \( L \). It shows a small section at a distance \( z \) from the center of the cylinder, labeled \( dz \). The center of the cylinder is marked \( O \). The figure helps in visualizing how a thin ring of width \( dz \) is considered at a distance \( z \) for calculating the electric potential at the center due to this ring.
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 figuring out part A, I was wondering if you can do part A STEP BY STEP so I can understand
![**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:
\[
V_{\text{center}} = \frac{1}{4\pi \varepsilon_0 L} Q \ln \left( \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}} \right).
\]
(The integral that you need to use here is \( \int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2+a^2}} = \ln(t + \sqrt{t^2+a^2}) \big|_{t_1}^{t_2} \).)
**Figure 2: The scheme for Problem 2**
The figure illustrates a hollow cylindrical shell. The cylinder has a radius \( R \) and a length \( L \). It shows a small section at a distance \( z \) from the center of the cylinder, labeled \( dz \). The center of the cylinder is marked \( O \). The figure helps in visualizing how a thin ring of width \( dz \) is considered at a distance \( z \) for calculating the electric potential at the center due to this ring.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F31a7d002-7306-48cb-8411-4964e654c6a5%2F32b5208e-cb9c-49e0-ba73-014e98bca3f9%2Fuwb561w_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:
\[
V_{\text{center}} = \frac{1}{4\pi \varepsilon_0 L} Q \ln \left( \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}} \right).
\]
(The integral that you need to use here is \( \int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2+a^2}} = \ln(t + \sqrt{t^2+a^2}) \big|_{t_1}^{t_2} \).)
**Figure 2: The scheme for Problem 2**
The figure illustrates a hollow cylindrical shell. The cylinder has a radius \( R \) and a length \( L \). It shows a small section at a distance \( z \) from the center of the cylinder, labeled \( dz \). The center of the cylinder is marked \( O \). The figure helps in visualizing how a thin ring of width \( dz \) is considered at a distance \( z \) for calculating the electric potential at the center due to this ring.
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