AP Free Response Question 2000 C3 E&M
A capacitor consists of two conducting, coaxial, cylindrical shells of radius a and b, respectively and length L >> b. The space between the cylinders is filled with oil that has a dielectric constant . Initially, both cylinders are unchanged, but then a battery is used to charge the capacitor, leaving a charge of +Q on the inner cylinder and -Q on the outer cylinder, as shown below. Let r be the radial distance from the axis of the capacitor.

(a) Using Gauss's law, determine the electric field midway along the length of the cylinder for the following values of r, in terms of given quantities and fundamental constants. Assume end effects are negligible.

 i. a < r < b

 ii. b < r << L

(b) Determine the following in terms of the given quantities and fundamental constants.

 i. The potential across the capacitor

 ii. The capacitance of the capacitor

Now the capacitor is discharged and the oil is drained from it. As shown below, a battery of emf ε is connected to the opposite ends of the inner cylinder and a battery of emf is connected to opposite ends of the outer cylinder. Each cylinder has a resistance R. Assume that end effects and contributions to the magnetic field from the wire are negligible.

(c) Using Ampere's law, determine the magnitude B of the magnetic field midway along the length of the cylinders due to the current in the cylinders for the following values of r.

 i. a < r < b

 ii. b < r << L