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**Unformatted text preview: **Name: 32] W“ D: I Student ID number: _
- Physics 17, Fall 2016
Midterm 1, October 21, 2016 (Friday) Version A READ THE FOLLOWING CAREFULLY o This exam is closed book. One 3"x5" index card with notes is allowed. Calculators are not
allowed. 0 This exam consists of 5 pages (including this one) with problems numbered 1 through 4; make
sure you have been given all pages/problems. - You have 50 minutes to complete the exam. 0 Make sure to write your name at the top of each page of this exam. Use the space provided on
the exam pages to do your work. You may use the back of the pages also, but please mark
clearly which problem you are working on (and also state underneath that problem that you
have done work on the back of the page). 0 . Partial credit will be given. Show as much work/justification as possible (diagrams where
appropriate). If you cannot figure out how to complete a particular computation, a written
statement of the concepts involved and qualitative comments on what you think the answer
should be may be assigned partial credit. 0 Mistakes in grading: If you find a mistake in the grading of your exam, alert the instructor within
one day of the exams being returned (this will occur in discussion section following the exam
date). DO NOT write on the returned graded exam — you may make a note of the problems you
thought were mis-graded on this ﬁrst page, but any changes/additions to the subsequent pages
will negate your chances for a re-grade and may result in the incident being reported to the
clean of students. A fraction of the graded exams will be photocopied before they are returned. Name: 1. [40 points] Density of modes: The essentials of calculating the number of modes of vibration of waves
confined to a cavity may be understood by considering a one—dimensional example. (3) Calculate the number of modes (standing waves of different wavelength A) per unit wavelength per
unit length that can exist on a string with fixed ends that is a length L long. Or, expressed
mathematically, calculate (1/L) dn/d7t. (b) Calculate the number of modes per unit frequency f per unit length that can exist on a string with
fixed ends that is a length L long. Take the wave velocity on the string to be v. (c) A guitar string is 1 m long and has adjacent harmonics at 1200 and 1600 Hz. What is the speed of
the wave on this string? (d) Now consider a cubic cavity with sides of length L and conducting walls. Find an expression for the density of electromagnetic modes per unit wavelength it per unit volume. Explain any special factors
resulting from the underlined word in the previous sentence. «9 mozyx 3mm): 0 @ xsa 0%sz ﬁk L We? . . L. L ’TLU £65” 0L 2
if?! gTrCZ—L‘l‘ a k 5M5 ‘lmq—llch‘O/‘WAS‘ Name: 2. An electron of momentump is at a distance r from a stationary proton. The system has a kinetic
energy K=p2/2me and potential energy U= - ke2/r. Its total energy is E = K + U. If the electron is bound
to the proton to form a hydrogen atom, its average position is at the proton. Here we take the
uncertainty in its position to be equal to the diameter 2 r, of its orbit. The electron’s average
momentum will be zero, but the uncertainty in its momentum will be given by the uncertainty principle.
Treat the atom as a one~dimensional system in the following: (a) Estimate the uncertainty in the electron’s momentum in terms of r. (b) Estimate the electron’s kinetic, potential, and total energies in terms of r. (c) The actual value of r is the one that minimizes the total energy, resulting in a stable atom. Find that
value of r and the resulting total energy. Compare your answer‘with the predictions of the Bohr theory. V5 gain/7 163(40me 0< $319133 is 49% ESQ ”game; 3 Dasha-Jammacllsm Aw ¢J% Jyoth Name: ~——-—-——————_u_________ ' r n ”0/ ‘1', if
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temperature T. You may leave your answer in terms of known constants. (A problem very like this one
was a homework question — the ﬁgure below is included to help remind you. You will find a
transcendental equation that you cannot solve exactly. Manipulate the equation so that it contains one exponential of the form 6". You will see that x is reasonably la rge, so you can neglect this term and
thereby arrive at a solvable equation and thus an approximate expression for fmax-) 12—]- g 5 0 mi, km” W W); oCC 0/5 LUZ/m M1.— O +} Inn-min prr mil murlrnmh A! l o, ‘ _. , 'Erl: . m. i I: ” KT Ema—a
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l<’l‘ ' On LCM: afr— +3 F. Name: 4. The energy-momentum relation for a free relativistic electron is E2=p2cz+m2c" W‘mf S- a) Considering the electron as a quantum-mechanical wave, re-write this expression in terms of the
electron’s angular frequency to and its wavenumber k. [0 b) Give expressions for the phase velocity vp and group velocity vg of this wave in terms of a) and k.
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- Physics, mechanics, Statistical Mechanics, Total Energy, student ID number, partial credit, unit wavelength, free relativistic electron