Phys 312 - Assignment 4 - Due 12 Feb 98

This assignment contains many short questions and a long one. Several questions are taken from Bloomfield, Section 11.2, with some changes.

1. (2 points)

It is possible to combine the fundamental constants $\hbar$ and e (elementary charge) to obtain a quantity with the dimensions of conductance. Using dimensional analysis, find this ``fundamental conductance'' and compare it with that of one meter of copper wire having a 1 mm radius, at room temperature. Recall that conductance is defined as the inverse of resistance, 1/R. Do not confuse conductance with conductivity, or resistance with resistivity.

2. (1 point)

A current of 1 Ampere flows in a copper wire. In how many seconds will a billion electrons pass through a given point on the wire?

3. (4 points)

Thermal energy can shift some of the electrons in a hot semiconductor from valence levels to conduction levels. What effect do these shifts have on the resistivity of the semiconductor? In an intrinsic semiconductor or insulator, the resistivity $\rho$ is found to change by a factor f when the temperature T is reduced from 300K to 150K. Is f greater or smaller than 1? What happens to $\rho$ when T is reduced to 75K? If the band gap is 1.4 eV, compute $\rho _{150}/\rho _{300}$ and $\rho _{75}/\rho _{300}$.

4. (2 points)

Based on the result of problem 3, would heating the photoconductor in a xerographic copier improve or diminish its ability to produce sharp, high-contrast images?

5. (2 points)

What is the maximum energy of a photon of visible light? How does it compare with the typical band gap of (a) insulators and (b) semiconductors? How does this explain the fact that typical semiconductors are opaque, while insulators can be clear or translucent?

6. (2 points)

Actually, good insulators can be opaque. Some, like dry wood or rubber, consist of large molecules. Can you explain, qualitatively, how a solid consisting of large molecules, or more generally with a complex structure, can be both opaque and insulating?

7. (extra points given for good long answers )

(a) (1 point) Why do incandescent lamps often burn out just after they are switched on?
(b) (1 point) Why do old bulbs show a dark spot at the top (rather than somewhere on the side)? See Bloomfield , section 6-3.

8. (10 points and possibly extra points)

(a) Plot the potential V=x²/(1+x⁵/100) for x between -2 and +10 and find its maximum value inside this interval (endpoints excluded).
(b) Integrate numerically the Schrödinger equation

$$-\frac{d^{2}y}{dx^{2}}+Vy=Ey$$

for several values of E between 0 and the maximum of V. Start the integration at x = -2, assuming that the total potential is infinite for x<-2. Find the approximate ground state and comment on the tunneling probability.