$(30$ pts$.)$ A long metal pipe of circular cross$-$section and radius $R$ is divided into four equal sections $($see the picture below$).$ Three of them are at constant potential $V_0$ and the fourth section is grounded. Use cylindrical coordinates to find

a$)$ Electric potential inside of the pipe,

b$)$ Electric potential outside of the pipe,

c$)$ Electric field $($both components$)$ inside of the pipe,

d$)$ Electric field $($both components$)$ outside of the pipe,

e$)$ Surface density of the electric charge on the inner and on the outer surfaces of the pipe,