E&M, Fall '10, Homework 8

1. A circuit is made out of a battery of EMF "epsilon", two identical capacitors of capacitance C, and a resistor of resistance R. The electrons flow out of one end of the battery, "through" one capacitor, then they split and some of them go "through" the other capacitor while some go through the resistor, then they all rejoin and hold hands while they re-enter the battery through the other terminal. That is: the circuit has a capacitor in series with a capacitor-and-resistor-in-parallel-combination. OK. Now, suppose at t=0 the capacitors are uncharged and the circuit is just hooked up. Draw *qualitative* graphs (that is, no equations or math) to capture what happens in the circuit. For example, draw a graph of the current through the battery (as a function of time), the charge on the two capacitors (...), and/or the current through the resistor (...).

2. What is a dielectric? Suppose a slab of dielectric material is slid in between the plates of a charged capacitor (that is disconnected from any circuits). What happens? Does the charge on the capacitor change? Does the potential difference across its terminals change? How can one understand all of this "microscopically"?

3. It has been claimed that Gauss' Law is equivalent to Coulomb's law (i.e., the inverse square law for how the electric field behaves in the vicinity of a point charge). See how far you can get in really establishing this equivalence. For example, can you show that Gauss' Law implies Coulomb's law? Can you show that Coulomb's law implies Gauss' Law? (Both are possible, though the second is a bit tricky; see how close you can get to a real proof.)

4. Use Gauss' Law to determine the electric field everywhere for a hollowed-out ball (inner radius "a", outer radius "b") with total charge Q uniformly distributed over the volume between "a" and "b". Now find the electric potential everywhere.