(Solved):     An infinitely long, hollow cylinder with inner radius \( a \) and outer radius \( b ...

   

An infinitely long, hollow cylinder with inner radius \( a \) and outer radius \( b \) carries a uniform positive charge density \( \rho \), and lies along the z-axis. Calculate the magnitude of the electric field that the cylinder generates inside the hollow part of the cylinder \( (rb) \) using Gauss's law. The cylinder remains unchanged when rotated around the \( z \)-axis or when shifted along the \( z \)-axis (cylindrical symmetry). The field must possess this same symmetry. This implies that the field can only point directly away from the z-axis at all points in space, and that its magnitude only depends on the distance to the \( z \)-axis. Motivate solutions clearly using full sentences and figures. (a) Choose a Gaussian surface as a cylinder with radius \( \mathrm{r} \) and length \( l \) which also lies along the zaxis. Express the electric flux \( d \Phi_{E}=E d A \) through a single surface patch on the cylinder in terms of the (unknown) field strength \( E(r) \). Consider two cases: (i) The patch lies on the curved part of the Gauss cylinder's surface. (ii) The patch lies on one of the flat ends of the Gauss cylinder. (b) Derive an expression for the total electric flux \( \Phi_{E}(r) \) through the Gaussian surface in terms of \( E(r) \). Show surface integrals clearly. (c) Write down an expression for the total charge \( Q_{\mathrm{enc}}(r) \) enclosed by the Gaussian surface. Consider three separate cases here. (d) Apply Gauss's law to the results of the previous two questions and write down expressions for \( E(r) \) in the three regions. (e) Draw a simple sketch of \( E(r) \) for \( r \) between 0 and \( 3 b \). Indicate any relevant values on the two axes.