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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?

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I keep getting the wrong answer for all of these problems and I don't know what I am doing wrong, can you show me how you did part a, part b and part c because I need help with those three problems. 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 \[ V_{\text{center}} = \frac{1}{4\pi \epsilon_0 L} Q \ln \left( \frac{\sqrt{R^2 + \frac{L^2}{4} + \frac{L}{2}}}{\sqrt{R^2 + \frac{L^2}{4} - \frac{L}{2}}} \right) \] (The integral that you need to use here is \( \int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2 + a^2}} = \ln \left( t + \sqrt{t^2 + a^2} \right) \Bigg|_{t_1}^{t_2} \)). **Figure 2: The Scheme for Problem 2** The diagram illustrates a hollow cylindrical shell with radius \( R \) and length \( L \). The diagram references a thin ring of width \( dz \) at a distance \( z \) from the center \( O \) of the cylinder along the z-axis. The integration process involves summing contributions from all such rings along the length of the cylinder.
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 \[ V_{\text{center}} = \frac{1}{4\pi \epsilon_0 L} Q \ln \left( \frac{\sqrt{R^2 + \frac{L^2}{4} + \frac{L}{2}}}{\sqrt{R^2 + \frac{L^2}{4} - \frac{L}{2}}} \right) \] (The integral that you need to use here is \( \int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2 + a^2}} = \ln \left( t + \sqrt{t^2 + a^2} \right) \Bigg|_{t_1}^{t_2} \)). **Figure 2: The Scheme for Problem 2** The diagram illustrates a hollow cylindrical shell with radius \( R \) and length \( L \). The diagram references a thin ring of width \( dz \) at a distance \( z \) from the center \( O \) of the cylinder along the z-axis. The integration process involves summing contributions from all such rings along the length of the cylinder.
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