A uniform volume charge density of \(0.2 \mu \mathrm{C} / \mathrm{m}^{3}\) is present throughout the spherical shell extending from \(r=3 \mathrm{~cm}\) to \(r=5 \mathrm{~cm}\). If \(\rho_{v}=0\) elsewhere, find \((a)\) the total charge present throughout the shell, and \((b) r_{1}\) if half the total charge is located in the region \(3 \mathrm{~cm}

a) Calculate the total charge present throughout the shell. Using the volume formula for spheres and subtracting the smaller sphere from the larger sphere, we find the volume of the shell: \(V_{shell} = \frac{4}{3}\pi {(0.05\ m)}^3 - \frac{4}{3}\pi {(0.03\ m)}^3\). Next, we use the given volume charge density (\(\rho_v = 0.2 \mu C/m^3\)) to determine the total charge in the shell: \(Q_{total} = \rho_v \cdot V_{shell}\). b) Determine the value of \(r_1\) where half of the total charge is located in the region \(3\text{ cm} < r < r_1\). We set up the equation using the charge density and the volume of this region: \(\frac{1}{2} Q_{total} = \rho_v \cdot (V_{r1} - V_{small})\). After substituting the values for \(Q_{total}\), \(V_{small}\), and \(\rho_v\), we solve the equation for \(r_1\), giving us the radius where half of the total charge is located.