Physics 3221 Fall Term 2004 Test 1, October 1, 2004

Physics 3221
Fall Term 2004
Test 1, October 1, 2004
This is an open notes test lasting 50 minutes.
There are two problems, divided into subsections. Problem 1 has 2 parts
and problem 2 has 4 parts. The points for each part are marked. Bold
face letters denote vectors.
Begin each problem on a fresh sheet of paper. Use only one side of a
sheet of paper.
Put your name, the problem number, and the page number in the upper
right hand corner of each sheet.
To receive partial credit you must explain what you are doing. Carefully
labeled figures are important. Randomly scrawled equations aren't helpful.
Draw a box around important results.
Some constants which may be useful:
Permittivity of free space ε0 = 8.85 × 10-12 C2/N.m2
electronic charge e = 1.6 × 10-19 C
electronic mass m = 9.1 × 10-31 kg
Boltzmann’s constant kB = 1.38 × 10-23 J/K
There are 2 pages including this page. Do not forget to look at all parts of the
Problem 1.
(a) The conductivity (σ) of a material is the inverse of its resistivity (ρ) (resistivity is
defined using the formula R = ρl/A). The σ of a metal depends on the number of
free electrons per unit volume (n), the electronic charge (e), the electronic mass
(m) and a parameter τ which denotes the time between two collisions of an
electron with other objects in the metal such as impurities. Using dimensional
analysis express σ in terms of n, e, m and τ. (6 points)
(b) For the metal Lithium, the value of resistivity (ρ) at 77 K is 1.04 µΩ-cm and the
value at 273 K is 8.55 µΩ-cm. If the τ at 77 K is 7.3 × 10-14 seconds, use your
result from (a) to find the value of τ at 273 K. (2 points)
Problem 2.
(a) Let R be the distance from a fixed point A with coordinates (a,b,c) to any point P
with coordinates (x,y,z). Show that ∇R is a unit vector in the direction AP = R. (5
(b) Let P be any point on an ellipse whose foci are at points A and B as shown in the
figure. The sum of the distances from A to any point P on the ellipse and from B
to point P on the ellipse is a constant (AP + BP = constant). Using this property
write down the equation of the ellipse in terms of R1 and R2. (2 points)
(c) Find a normal vector to the ellipse at point P using the equation from part (b). (2
points) [Hint: Use the necessary property of the gradient operator that we
worked out in class, only in this case there are only 2-dimensions. You can
keep your answer in terms of R1 and R2.]
(d) Using the results from part (a) and (c) show that R1 and R2 make the same angle
to the tangent vector T at point P (i.e. θ1 = θ2.) (3 points)