**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) Surface Charge Density:** Compute the surface charge density \( \eta \) of the shell from its total charge and geometrical parameters. **b) Charge in a Thin Ring:** 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 (using the formula derived in the textbook for the potential of a charged ring along its axis)? **c) Integration:** Sum up the contributions from all the rings along the cylinder by integrating \( dV \) with respect to \( z \). Show that: \[ V_{\text{center}} = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{L} \ln \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}} \] (The integral 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 Description:** The diagram (Fig. 2) illustrates a hollow cylindrical shell with radius \( R \), length \( L \), and a thin ring of width \( dz \) at a distance \( z \) from the center of the cylinder. The diagram depicts the central axis and points out the relevant parameters used in the problem.
**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) Surface Charge Density:** Compute the surface charge density \( \eta \) of the shell from its total charge and geometrical parameters. **b) Charge in a Thin Ring:** 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 (using the formula derived in the textbook for the potential of a charged ring along its axis)? **c) Integration:** Sum up the contributions from all the rings along the cylinder by integrating \( dV \) with respect to \( z \). Show that: \[ V_{\text{center}} = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{L} \ln \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}} \] (The integral 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 Description:** The diagram (Fig. 2) illustrates a hollow cylindrical shell with radius \( R \), length \( L \), and a thin ring of width \( dz \) at a distance \( z \) from the center of the cylinder. The diagram depicts the central axis and points out the relevant parameters used in the problem.
Related questions
Question
Hello, I really need help with Part A, PART B AND PART C because I have no idea how to do this problem and I need help because I keep getting the wrong answer. I was wondering if you can help me with PART A, PART B AND C and can you also label them as well.
![**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) Surface Charge Density:**
Compute the surface charge density \( \eta \) of the shell from its total charge and geometrical parameters.
**b) Charge in a Thin Ring:**
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 (using the formula derived in the textbook for the potential of a charged ring along its axis)?
**c) Integration:**
Sum up the contributions from all the rings along the cylinder by integrating \( dV \) with respect to \( z \). Show that:
\[
V_{\text{center}} = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{L} \ln \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}}
\]
(The integral 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 Description:**
The diagram (Fig. 2) illustrates a hollow cylindrical shell with radius \( R \), length \( L \), and a thin ring of width \( dz \) at a distance \( z \) from the center of the cylinder. The diagram depicts the central axis and points out the relevant parameters used in the problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd2a66187-0c81-42c5-b98f-d9dd1e72eaec%2F0c72a361-db24-4253-bac7-64b69b81db47%2Fsylpixr_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) Surface Charge Density:**
Compute the surface charge density \( \eta \) of the shell from its total charge and geometrical parameters.
**b) Charge in a Thin Ring:**
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 (using the formula derived in the textbook for the potential of a charged ring along its axis)?
**c) Integration:**
Sum up the contributions from all the rings along the cylinder by integrating \( dV \) with respect to \( z \). Show that:
\[
V_{\text{center}} = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{L} \ln \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}}
\]
(The integral 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 Description:**
The diagram (Fig. 2) illustrates a hollow cylindrical shell with radius \( R \), length \( L \), and a thin ring of width \( dz \) at a distance \( z \) from the center of the cylinder. The diagram depicts the central axis and points out the relevant parameters used in the problem.
Expert Solution
Step 1
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
