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Two infinitely long, cylindrical conducting shells are concentrically arranged as shown above. The inner shell has a radius $R_1$ and the outer shell the radius $R_2$. The inner shell has a charge per unit area $\sigma_1$, and the outer shell a charge per unit area $\sigma_2$.

a) Find the electric field $\vec{E}$ at all points in space (you should have three answers for three distinct regions).

b) Find the surface charge density $\sigma_2$ (in terms of $\sigma_1, R_1, R_2$, etc.) that causes the field to vanish everywhere but in between the two shells. Justify your answer with Gauss's law.