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Practice Final (b) - 1Bb Physics for Scientists and...

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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 —— _— _— —— ' 5- _ _ 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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