**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 \(\int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2 + a^2}} = \ln(t + \sqrt{t^2 + a^2}) \Big|_{t_1}^{t_2}\) ) --- **Figure 2: The scheme for Problem 2** The diagram illustrates a hollow cylindrical shell with a uniform distribution of charge \( Q \) across its length. The cylinder has a length \( L \) and radius \( R \). The electric potential at the center of the cylinder is to be calculated by considering the contribution from each differential segment \( dz \), which forms a ring at a distance \( z \) from the center.

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 \(\int_{t_1}^{t_2} \frac{dt}{\sqrt{t^2 + a^2}} = \ln(t + \sqrt{t^2 + a^2}) \Big|_{t_1}^{t_2}\) ) --- Figure 2: The scheme for Problem 2 The diagram illustrates a hollow cylindrical shell with a uniform distribution of charge \( Q \) across its length. The cylinder has a length \( L \) and radius \( R \). The electric potential at the center of the cylinder is to be calculated by considering the contribution from each differential segment \( dz \), which forms a ring at a distance \( z \) from the center.