Selected Problems Tutorial # 2

1. One end of an Al wire (diameter 2.5mm) is welded to one end of a Cu wire (diameter 1.8mm). The composite wire carries current I=1.3A. What is the current density in each wire?

Answer: Cross-sectional area A of Al wire Current density (constant within wire except near the junction) For copper: A Cu = 2.54 x m 2 J Cu = 5.1 x 10 5 A/m 2

2. What is the drift speed of the conduction electrons in the copper wire of Problem 1? Answer: In Cu, there is nearly 1 conduction electron per atom (on the average). n = number of electrons per unit volume equals the number of atoms per unit volume Avogadro’s number: N A = 6.02 x /mol density of copper: = 9.0 x 10 3 kg/m 3 molar mass of copper: M = 64 x kg/mol

n = 8.47 x electrons/m 3 = 3.8 x m/s = 14 cm/s Electrons drift very slowly.

3. A strip of silicon has a width of 3.2mm and a thickness of 250 m, and it carries a current I=5.2mA. The silicon is an n-type semiconductor, having been “doped” with a controlled phosphorus impurity. The doping has the effect of greatly increasing n, the number of charge carries per unit volume, as compared with the value for pure silicon. In this case, n=1.5 x m -3.

Q: a) What is the current density in the strip? Answer: w=width t=thickness

Q: b) what is the drift speed? Answer: = 0.27m/s = 27cm/s Note: The drift speed (0.27m/s) calculated for the electrons in this doped silicon semiconductor is much greater than the drift speed (3.8 x m/s) obtained in Problem 2 for the conduction electrons of the metallic copper conductor.

4. What is the strength of the electric field present in: a) a copper conductor in which the current density is Answer: Resistivity of copper

b) an n-type silicon semiconductor in which the current density is J=6500A/m 2. Answer: Resistivity of n-type silicon semiconductor =5.7V/m

c) comment on the results obtained in a) and b). Answer: The electric field in the silicon semiconductor (5.7V/m) is considerably higher than that in the copper conductor (8.6 x V/m). This is due to the much lower concentration of charge carriers in silicon than in copper. For a given current density, the fewer charge carriers in silicon must drift faster, which means that the electric field acting on them must be stronger.

5. a) What is the mean free time between collisions for the conduction electrons in copper? A Answer: n=8.47 x electrons/m 3 (see example problems) where units were converted as

b) What is the mean free path for these collisions? Assume an effective speed v eff of 1.6 x 10 6 m/s. Answer: =4.0 x m = 40nm This is about 150 times the distance between nearest neighbor ions in a copper lattice.