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Unformatted text preview: 1Bb Physics for Scientists and Engineers: Oscillations, Waves, Electric
ity and Magnetism Final b. Thursday 15th June. Instructor: Steve Cowley ——
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_ Total. Answer 4 out of 5 questions — indicate above which questions you want us to mark. Open book, open notes ~ no talking. Please write your answers in the space provided below. Partial credit is
given for answers that are on the right track so show your working. You may use rough paper but please put the
working on these sheets. Answers need not be more accurate than two significant ﬁgures. If you don’t have much
time you will get almost all of the credit for writing down the right expression (with the right numbers inserted)
and leaving the arithmetic undone. No calculators or cell phones are to be used in the exam. Question 1. On Reﬂection. 20 points
We consider sound waves propagating in air  the sound speed is labeled 1), and it has the value 118 = 345ms‘1.
The oscillating air velocity u(:t:, t) obeys the wave equation:
a: r 2:
(9t2 — 3 8x2 (a) Write down the traveling wave solution (for u(x,t)) for a wave of wavelength 27rm and amplitude 3ms’1 that is traveling towards as = 00? 5 points (b) A wall that reﬂects the wave is placed at m = 0 — write down the form of u(:1:,t) including the reﬂected
wave and the incident wave from part (a). Find the amplitude and phase of the reﬂected wave given that at the wall u(0, t) = 0 5 points (c) The wall is now moved at velocity vwa” in the positive x direction so that the position of the wall is
w = mum” = vwaut. Calculate the frequency and wavelength of the reﬂected wave when vwa” = 172.5ms‘1. 4 points ((1) Apply the reﬂection condition that Mama”, t) = 0 on the moving wall to determine the amplitude and phase
of the reﬂected wave. 4 points (e) Describe the reﬂected wave when the wall is oscillated with a small amplitude at a frequency fwa”. 2 points Question 2. Coaxial but Different. 20 points
Consider a coaxial cable made from two thin tubes of
radius a. A total current of I ﬂows down the inner tube an conductor. The outer tube of radius b and the inner of
d a return current of ——I ﬂows down(up) the outer tube. UMJ‘WNLW“ (a) From symmetry arguments write down a general form of the magnetic ﬁeld, i.e. it’s direction and how it depends on the coordinates. You don’t have to actually solve for it yet . 4 points (b) Find the magnetic ﬁeld in the three regions: 7' < a, a < 1‘ < b and b < 1‘. 6 points (c) Suppose that the outer tube also has a current ﬂowing in the 0 direction of magnitude Igamps/m per unit
length. Note this means that in a meter length of tube the total current in the 0 direction is I9. Recalculate the ﬁeld in the three regions. 6 points (d) Consider the case without the 19. What are the forces on the inner and outer tubes — give the directions
and if you can the magnitude. 4 points Question 3. RC Circuit. 20 points
Consider a circuit with two resistors and a capacitor (see picture). Vmc
W (a) Apply Kirchoff’s voltage law to the circuit and relate the current and the charge on the capacitor. Write
down a differential equation for the charge in the capacitor. 4 points (b) Find the solution for the charge as a function of time when the charge is zero at time t = 0. 6 points (c) After the capacitor has charged we disconnect the battery and connect another (identical) but uncharged
capacitor in its place. Label the charges on the capacitors as shown. Write down Kirchoff’s laws for the new circuit
and the relationship between I and the charges. Calculate the subsequent evolution of the charges — to make things
easier you may relabel time so that t = 0 corresponds to the moment the second capacitor is connected to the circuit. 8 points (d) What would be the ﬁnal state if the second capacitor had the capacitance of C / 2.. 2 points Question 4. Electric Fields in Capacitors. 20 points
Consider a capacitor of area A and separation L.
3’ “till: H45” (a) Find the charge on the capacitor when a voltage V is applied to the two plates. 4 points (b) A conducting sheet of thickness L / 2, the same area with no net charge is placed between the two plates.
Find the charge on the surfaces of the plates and the surfaces of the conductor in this case. 4 points (c) The conducting sheet is replaced with a dielectric sheet of dielectric constant It. Find the charge on the
surfaces of the plates and the dielectric in this case.4 points (d) Find the energy of the capacitor with and without the dielectric when a voltage V is applied. 4 points (e) The dielectric sheet is pulled half out of the capacitor (see picture). Will it be pulled inwards or outwards
by the electrical forces? Hint, an enemy argument is helpful.4 points Question 5. Driven Oscillators. 20 points
Consider the driven oscillations of a car in a suspension test. The car position m(t) obeys the equation. 2 d
:7: +11% +w3x =c0th (a) Write down the solution for long times — i.e. after the initial transients. 4 points (b) Find the longtime solution when 0 = we = 13“1 and 1/ = 0.1. 4 points (c) The force is modulated so that d2 d1: 2
872 + 11% + wow 2 coswltcoswot Find the longtime solution when wl = 11/2 simplify this expression for small 1/. 4 points (d) The force (the right hand side of the equation) in part (a) is turned off at time t = 2007rs. Sketch and
calculate the subsequent motion. 4 points (e) Recall that the damping force is —mu‘fi—f Where m is the mass of the car. Find an expression for the power
expended against this force. Calculate the average power for the parameters of part (b). 4 points ...
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