1985E1. A capacitor consisting of conducting coaxial cylinders of radii a and b, respectively, and length L is connected to a source of emf, as shown above. When the capacitor is charged, the inner cylinder has a charge + Q on it. Neglect end effects and assume that the region between the cylinders is filled with air. Express your answers in terms of the given quantities.
a. Use Gauss's law to determine an expression for the electric field at a distance r from the axis of the cylinder where a < r < b.
b. Determine the potential difference between the cylinders.
c. Determine the capacitance Co of the capacitor.
One third of the length of the capacitor is then filled with a dielectric of dielectric constant k = 2, as shown in the following diagram.
d. Determine the new capacitance C in terms of Co.
1995E2. A parallel‑plate capacitor is made from two sheets of metal, each with an area of 1.0 square meter, separated by a sheet of plastic 1.0 millimeter (10‑3 m) thick, as shown above. The capacitance is measured to be 0.05 microfarad (5 x 10‑8 F) .
a. What is the dielectric constant of the plastic?
b. The uncharged capacitor is connected in series with a resistor R = 2 x 106 ohms, a 30‑volt battery, and an open switch S, as shown above. The switch is then closed.
i. What is the initial charging current when the switch S is closed?
ii. What is the time constant for this circuit?
iii. Determine the magnitude and sign of the final charge on the bottom plate of the fully charged capacitor.
iv. How much electrical energy is stored in the fully charged capacitor?
After the capacitor is fully charged, it is carefully disconnected, leaving the charged capacitor isolated in space. The plastic sheet is then removed from between the metal plates. The metal plates retain their original separation of 1.0 millimeter.
c. What is the new voltage across the plates?
d. If there is now more energy stored in the capacitor, where did it come from? If there is now less energy, what happened to it?
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