A very long conducting tube (hollow cylinder) has inner radius aa and outer radius b. It carries charge per unit length +α, where αα is a positive constant with units of C/m. A line of charge lies along the axis of the tube. The line of charge has charge per unit length +α. Calculate the electric field in terms of α and the distance r from the axis of the tube for r<a. Calculate the electric field in terms of α and the distance rr from the axis of the tube for a<r<b. Calculate the electric field in terms of αα and the distance r from the axis of the tube for r>b. What is the charge per unit length on the inner surface of the tube? What is the charge per unit length on the outer surface of the tube?

Two small insulating spheres with radius 9.00×10−2 m are separated by a large center-to-center distance of 0.545 m . One sphere is negatively charged, with net charge -1.75 μC , and the other sphere is positively charged, with net charge 3.70 μC . The charge is uniformly distributed within the volume of each sphere.  What is the magnitude E of the electric field midway between the spheres? Take the permittivity of free space to be ϵ0 = 8.85×10−12 C2/(N⋅m2) .  What is the direction of the electric field midway between the spheres?

A conducting spherical shell with inner radius aa and outer radius bb has a positive point charge Q located at its center. The total charge on the shell is -3Q, and it is insulated from its surroundings.  Derive the expression for the electric field magnitude in terms of the distance r from the center for the region r<a. Express your answer in terms of some or all of the variables Q, a, b, and appropriate constants. Derive the expression for the electric field magnitude in terms of the distance rr from the center for the region a<r<b. Derive the expression for the electric field magnitude in terms of the distance rr from the center for the region r>b. What is the surface charge density on the inner surface of the conducting shell? What is the surface charge density on the outer surface of the conducting shell?