Homework on Conductors and Semiconductors (solutions)

Feel free to work on the homework in groups. The work you hand in, however, should reflect your understanding of the material and be in your own words.Students who turn in identical (or close to identical) homework assignments will be asked to explain their answers orally to the TA or prof.  A student who cannot explain how he or she arrived at a given answer will be charged with academic dishonesty.

You should show all of your calculations (neatly) and justify all of your answers for full credit.
 

1.	Is current always directly proportional to voltage?  Explain, using the terms "ohmic" and "resistance". No.  In non-ohmic materials, resistance (defined as the ratio of voltage to current) is not constant.  Only in ohmic materials are current and voltage directly proportional.
2.	The resistances of various resistors are compared.  Resistor 1 is copper (a conductor) wire and has a resistance of 20 W at room temperature.  Resistor 2 is silicon (a semiconductor) and has a resistance of 30W at room temperature.  What is the resistance of the following resistors:  Resistor A, which is made from copper wire twice as long and the same gauge (i.e., has same radius) as Resistor 1, Resistor B, which is made from silicon the same length but with twice the radius of Resistor 2, and Resistor C, which is made from copper wire twice as long and with twice the radius of Resistor 2? Which material (silicon or copper) is resistor D, which has a resistance of 20 W at room temperature, and a resistance of 30 W when cooled? Resistance varies proportionally with length, so Resistor A has twice the resistance of Resistor 1, or 40 W. Resistance varies inversely with area, so Resistor B has 1/4 the resistance of Resistor 2, or 7.5 W. The resistance of Resistor C is unknown, because a typo from last year was copied over again by Prof. Schowalter.  Assuming that it should refer to Resistor 1, then it has half the resistance; double for length and 1/4 for radius, giving a result of 10 W. Resistor D is made out of silicon, the semiconductor, since resistance varies inversely with temperature.
3.	The temperature (T)-dependence of resistivity r is often described by r0 - r = a(T0 - T), where the subscript 0 represents the initial temperature and resistivity, and a is a material-dependent parameter.  Do you expect a to be positive, negative, or zero for a semiconductor?  for a conductor?  for an insulator?  Use band structure differences to explain your answer. For semiconductors, a is negative and fairly high.  Thermal energy is highly necessary for electrons to reach the conduction band from the full valence band, thus temperature will reduce resistance. For conductors, a is positive and fairly small.  Electrons have space to move in the valence band, and thus current can flow at nearly any temperature.  Increasing the number of electrons able to move around only increases the number of collisions between them, which increases the resistance a bit. For insulators, a is very close to zero (perhaps slightly negative).  The electrons are bound very tightly, and an enormous amount of energy is required to reach the conduction band.  Unless the material is extraordinarily hot (usually enough to destroy the material), thermal energy will not suffice, and the resistance will remain essentially unchanged.