Formulas and Problems in Physics: Waves and Electromagnetism, Exams of Physics

Download Formulas and Problems in Physics: Waves and Electromagnetism and more Exams Physics in PDF only on Docsity! Physics 171.201 Final Exam December 19th, 2006 Answer all six problems. Be sure that you pace yourself so that you have enough time to work on each problem. Note that the problems do not have equal weight. Partial credit will be given, so be sure to show your work as clearly as possible. Good luck! List of potentially useful formulae ! " x = x # vt 1# v 2 /c 2 " y = y " z = z " t = t # (v /c 2 )x 1# v 2 /c 2 ux " = ux # v 1# uxv /c 2 uy " = uy 1# v 2 /c 2 1# uxv /c 2 ! m = m0 1" v 2 /c 2 E = mc 2 r p = m r v E 2 = m0 2 c 4 + p 2 c 2 E 2 " p 2 c 2 = # E 2 " # p 2 c 2 Maxwell’s Equations ! " # r E = 0 " # r B = 0 ! " # r E = $1 c % r B %t " # r B = 1 c % r E %t Poynting Vector: ! r S = c 4" 1 µ r E # r B EM Energy Density: ! U = 1 8" # r E 2 + 1 µ r B 2 $ % & ' ( ) Snell’s Law: ! n1 sin("1) = n2 sin("2 ) Potentially useful formulae, continued: General differential equation for force, damped harmonic oscillator: ! ˙ ̇ x + "˙ x +#0 2 x = F0 cos(#t) has solutions, ! x(t) = Ae "#t /2 cos $0 2 " # 2 4 % & ' ( ) * 1/2 t + + % & ' ' ( ) * * + xss(t) $0 > # /2 (A+ Bt)e "#t /2 + xss(t) $0 = # /2 Ae ",1t + Be ",2t + xss(t) $0 < # /2 - . / / / / 0 / / / / where ! "1 = # 2 + #2 4 $%0 2 and ! "1 = # 2 $ #2 4 $%0 2 and the steady state solution is: ! xss(t) = A(")cos("t # $(")) where , ! A(") = F0 "0 2 #"2( ) 2 + $2"2 % & ' ( ) * 1/2 and ! tan("(#)) = $# #0 2 %#2 ________________________________________ The wave equation: ! " 2 "x 2 y(x,t) = 1 v 2 " 2 "t 2 y(x,t) where, e. g., ! v = T /" a string. Two dimensional wave equation: ! " 2 z "x 2 + " 2 z "y 2 = 1 v 2 " 2 x "t 2 __________________________________________ Reflection and transmission of EM waves: For E field: ! R = Z2 " Z1 Z2 + Z1 ! T = 2Z2 Z1 + Z2 where Z=1/n (n = index of refraction) For B field: ! R = Z1 " Z2 Z2 + Z1 ! T = 2Z1 Z1 + Z2 Problem 3 (30 points) The picture below on the left shows a simple pendulum comprised of a light rigid rod of length ! l supporting a mass M. (a) Show that if the angular displacement θ of the pendulum is small, then the equation of motion for the mass can be written in terms of the lateral displacement x as: ! ˙ ̇ x + g l x = 0. Consider now two identical such pendulums coupled together by a spring of stiffness K as shown below on the right. Call the displacement of the mass on the left x1 and that of the mass on the right x2. (b) Write down the coupled differential equations for the motion of each of the masses that describe small amplitude displacements from their equilibrium positions. (c) Determine the normal mode frequencies for the system. (d) Imagine that the two masses are initially at rest in equilibrium. At time t = 0, a bullet strikes the mass on the left imparting a sudden impulse that gives the mass a velocity v0. The displacements and velocities of the masses at the instant after the bullet strikes are then: ! x1(0) = 0 ˙ x 1(0) = v0 ! x2(0) = 0 ˙ x 2(0) = 0 Find the ratio between the amplitudes of the two normal modes that describes the subsequent motion of the masses. M K M ! l ! l x 1 x 2 M x " ! l Problem 4 (30 points) As part of its search for extrasolar planets, NASA discovers a planet that appears to be very much like earth orbiting a star 40 light years from our solar system. An expedition is planned to send astronauts to the planet. NASA would like the astronauts to age no more than 30 years during the journey. In this problem, neglect any issues related to the acceleration of the astronauts’ spaceship. NOTE: A light year is a unit of distance. It is how far light travels in one year. For example, a spaceship with a velocity of 0.5c will require 80 years to travel 40 light years. (a) At what velocity must the astronaut’s spaceship travel in earth’s reference frame so that the astronauts age 30 years during the journey? [Hint: Measured in years, the trip according to an observer on earth will take a time Δt = 40c/v. (b) According to the astronauts in the spaceship, what will be the distance of their journey? (c) In order to let people on earth keep track of their progress, the astronauts shine a light pulse back to earth once each year, as measured by the clocks on the spaceship. How often do the light pulses reach earth? (d) In order to keep in touch with the astronauts, the people on earth similarly shine a light pulse to the ship once each year, as measured by clocks on earth. How often do the light pulses reach the spaceship? (e) Exactly half way to the planet, two of the astronauts get homesick and set off in a space module to return to earth. According to the astronauts who remain on the space ship, the module travels at a velocity of ! 5 6 c in the direction toward earth. Find the total amount of time that the two astronauts will have been away according to people on earth. Problem 5 (30 points) Consider two infinite walls with perfect electrical conductivity. One wall is positioned in the x-y plane at z = 0, and the other wall is parallel to this wall and located at z = L. Recall from E&M that the component of an electric field parallel to a conducting surface must go to zero at the surface. (a) Write down expressions for the three lowest-frequency electromagnetic standing waves polarized along the x-direction that can be created in the space between the walls. What are the frequencies of these modes? (Assume the space between the walls is vacuum.) (b) Using your result from (a) and Maxwell’s Equations, determine the magnetic field for the lowest-frequency standing wave. (c) Again determine the magnetic field for the lowest-frequency standing wave, this time using your result from (a) along with the fact that a standing wave can be considered the superposition of two traveling waves. Hint: you might find the following trig identities helpful: ! sin(x ± y) = sin(x)cos(y) ± cos(x)sin(y) ! cos(x ± y) = cos(x)cos(y) m sin(x)sin(y) (d) For the lowest-frequency standing wave, the electromagnetic energy flux that passes through an imaginary planes at x = L/4 and x = L/2 as a function of time. (The units of energy flux are ! energy area " time .)