**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 \[ V_{\text{center}} = \frac{1}{4\pi \varepsilon_0 L} Q \ln \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}}. \] (The integral that you need to use here is \(\int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2+a^2}} = \ln(t + \sqrt{t^2+a^2}) \bigg|_{t_1}^{t_2}\)). **Explanation of Fig. 2**: The figure illustrates a hollow cylindrical shell with length \( L \) and radius \( R \). The charge is distributed uniformly along the length of the cylinder. A thin ring with width \( dz \) is depicted, positioned at a distance \( z \) from the center \( O \) of the cylinder. The diagram conveys the geometric parameters necessary for calculating the electric potential at the center of the cylinder due to the charge distribution.

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 \[ V_{\text{center}} = \frac{1}{4\pi \varepsilon_0 L} Q \ln \frac{\sqrt{R^2 + \frac{L^2}{4}} + \frac{L}{2}}{\sqrt{R^2 + \frac{L^2}{4}} - \frac{L}{2}}. \] (The integral that you need to use here is \(\int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2+a^2}} = \ln(t + \sqrt{t^2+a^2}) \bigg|_{t_1}^{t_2}\)). Explanation of Fig. 2: The figure illustrates a hollow cylindrical shell with length \( L \) and radius \( R \). The charge is distributed uniformly along the length of the cylinder. A thin ring with width \( dz \) is depicted, positioned at a distance \( z \) from the center \( O \) of the cylinder. The diagram conveys the geometric parameters necessary for calculating the electric potential at the center of the cylinder due to the charge distribution.