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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 problems. 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 points) (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) T P θ1 R1 A R2 B θ2