Name Solutions to Test 3 April 11, 2012

Transcription

1 Name Solutions to Test 3 April, This test consists of three parts. Please note that in parts II and III, Constant you can skip one question of those offered. 7 4 Tm/A LC Circuits L Circuits Possibly useful formulas: L LC AC Circuits t I Ie Q Q cost V V sin t EM waves t E I e I I sin t c Impedance S E XC C, XL L MS values LC Circuits Z XL XC V S c V rms tan X P S c L XC LC 4L I t L Q Qe cost I LC rms Inductor L Faraday s Law Ampere s Law N A Generator d d E U E d s d s I LI NAsin t dt dt Circles Triangles Spheres Cylinders Cones C A H A 4 V L V 3 L A 4 3 V 3 Alat L A L Part I: Multiple Choice [ points] For each question, choose the best answer ( points each). In the simple circuit illustrated at right, at what frequencies will the most power be transferred to the resistor? A) Low frequencies ) High frequencies C) A broad range of intermediate frequencies D) A narrow band of intermediate frequencies E) Insufficient information. Electricity is produced in commercial generators by A) Giant banks of chemical batteries ) Extracting electrons from atoms and pumping them into wires C) ubbing objects together to produce static charges D) Moving loops of wire near magnetic fields (or vice versa) E) Accelerating electrons using light waves lat

2 3. If an electromagnetic wave has frequency f = Hz, what is the angular frequency? A) s - ) s - C) 4 s - D) /() s E) /(4 ) s 4. Which two components work best if you want to create an oscillator? A) Inductor and capacitor ) Inductor and resistor C) esistor and capacitor D) Inductor and diode E) Capacitor and diode 5. If you put power into an LC circuit, the power gradually drains away. Which component is actually consuming the power? A) The resistor (only) ) The capacitor (only) C) The inductor (only) D) The capacitor and the resistor E) The inductor and the resistor 6. Which of the following types of electromagnetic radiation moves fastest in vacuum? A) Infrared ) adio C) X-rays D) Gamma rays E) None; it is a tie 7. In an LC circuit, where is most of the energy stored at any given time? A) In the capacitor ) In the inductor C) It is split evenly at all times, with 5% in each component D) It is constantly being passed back and forth between the two components E) It is impossible to answer this question without know the capacitance and/or the inductance 8. Why do we use step-up transformers to increase the voltage very high before we transport it across the country? A) It has to go to many customers who each get just a little bit of this voltage ) Some customers (like industry) require very high voltage C) Since power is voltage by current, it vastly increases the power being transmitted D) High voltage is less dangerous, so it is just a safety consideration E) Since it is at high voltage, there is much less current needed, which decreases the losses in transmission 9. A resistor with resistance = 3 is connected in series to a capacitor with impedance X C = 4. What is the total impedance? A) ) 4 C) 5 D) 7 E) 5. Which of the following is the correct relation between the wavelength and wave number k? A) k ) k C) k D) k E) k

3 Part II: Short answer [ points] Choose two of the following questions and give a short answer (-3 sentences).. Explain qualitatively Lenz s Law. Suppose I took a superconducting loop of wire out of a region of magnetic field. What does Lenz s Law tell us in this case? Lenz s Law says when you try to change the magnetic flux passing through a conducting loop, a current will arise in the loop to try to oppose that change. In particular, if you remove a superconducting loop of wire out of a region of magnetic field, a current will spontaneously appear in the loop of wire which recreates that magnetic flux. ecause a superconductor has no resistance, this current will persist indefinitely.. The graph at right shows the current (dashed curve) that results from putting an AC voltage (solid curve) through a single component (resistor, capacitor, inductor). Identify which one, and explain how you can tell. The current (dashed curve) rises and falls one-quarter cycle before the voltage (solid curve); that is, the current leads the voltage. A resistor has no phase shift, and an inductor causes the current to lag the voltage, so this must be a capacitor, which indeed does cause current to lead the voltage. 3. Place the following in order from highest frequency to lowest: blue, green, orange, red, violet, yellow. You may include additional colors if you wish, but you don t get extra credit. Listed at right, with highest frequency first, and extra colors squeezed in on the right. Violet lue Green Yellow Orange ed Indigo Turquoise Chartreuse Saffron Vermillion

4 Part III: Calculation: [6 points] Choose three of the following four questions and perform the indicated calculations ( points each) 4. In 87, a spaceship approaches neutron star EC5 which has a magnetic field of.53 7 T and rotates. times per second. The captain then places a circular loop of wire of radius 5 m with turns in it in the vicinity of this magnetic field, so at t =, the magnetic field is perpendicular to the loop. (a) Find the magnetic flux through the loop at t =. magnetic field EC5 Wire loop Magnetic flux is given by A, where A is the area vector perpendicular to the loop. This makes A parallel to, so we then have 7 3 A.53 T 5 m. Wb. The problem is ambiguous; this is the flux through a single loop. If you count the twelve 4 loops (relevant for part (d)), you would multiply this by to get.44 Wb. (b) Half a cycle later, the neutron star has rotated, so that the field is now reversed. What is the flux now? What time is it now? Since it makes rotations a second, it will be reversed 4 s later, or.47 s. The flux will be the same, but reversed. Counting the twelve loops, it is now 4.44 Wb. (c) Assume the magnetic field changed linearly between part (a) and part (b). What was the EMF generated in the loop of wire during this half-rotation? Faraday s Law says that the EMF generated is the negative of the time derivative of the magnetic field. If we assume the magnetic field changed linearly (it probably didn t), then we would have Wb.44 Wb d dt t.47 s magnetic field V. None of the numbers given in the problem are unrealistic for a real neutron star. Whether we could ever travel to such a star, and build a conductor that could stand up to this voltage, however, is more in doubt.

5 5. A 4. H inductor and a.5 k resistor are connected as shown at right, and 5. A of current is then fed through it until it attains a steady state. Assume the inductor has no resistance. (a) In a steady state, how much current flows through the inductor? The resistor? 5 A 5 A 4. H.5 k In a steady state, the inductor has unchanging current, and hence there is no EMF across it. This tells us the voltage across it is zero, and thus the current in the resistor is zero. The inductor carries all the current. I 5. A, I. A. L (b) At t =, the external current is suddenly switched off. What is the current through the inductor and resistor a moment later? What is the voltage across the resistor at this moment? Inductors resist changes in current, and hence the inductor will still carry 5. A through it. ecause this current has nowhere else to go, it will force its way through the resistor, and hence this is the current in the resistor as well. I I 5. A. L Since resistors are governed by the equation V I, the voltage across the resistor will now be V I 5. A.5 k 7.5 kv. (c) How long will it take until the current in the inductor has dropped to. A? have: t The current is given by the equation I Ie, where L. We therefore t. A 5. A e, t 5. A e 5,. A t ln 58.57, L H t s.7 ms. 5

6 6. A certain device requires AC voltage V 5. V, with a imum current I.4 A, but all that is available is house current with Vrms V. (a) What is the rms voltage and current required by the device? The rms voltage and current can simply be found by dividing the imum by, so we have V 5. V I.4 A 3.54 V,.99 A. Vrms Irms (b) A transformer with 5 turns on the secondary (output) side will be used to supply this current. How many turns should there be on the primary (input) side of the transformer? Assume the transformer is % efficient. For an ideal transformer, we have V V, N N NV NV, V N N V V 5 7 turns 3.54 V (c) What will be the rms current drawn from the primary source? ecause we assume our transformer is % efficient, the power in will equal the power out; that is, I V I V, where all quantities are rms values. This allows us to compute the current in: I V 3.54 V.99 A.9 A 9. ma. I V. V

7 7. A black spaceship in the shape of a cone with radius 6. m and height 8. m is being impacted by a light 8 beam with intensity S 5. W/m beamed directly at the spaceship s pointy end for a duration of. hour. (a) What is the imum intensity of the magnetic field and electric field from this wave? 8. m 6. m 6. m We solve for the magnetic field using the equation S S 8 5. W/m 4.88 c, 4 Tm/A TNs c 3 m/s Cm T, 3.5 T.5 mt. We then get the electric field strength from E c 3. m/s.5 T 6.4 V/m. (b) What is the total momentum transferred to the spacecraft by the light beam? To get this right, we have to think carefully about the concept of cross-section. The cross-sectional area is the area as viewed by the incoming light ray. For this light ray, the cone is a circle of radius 6. m, so the cross sectional area is 6. m 3 m. We now multiply this by the pressure to get the force: 8 S 5. W/m 3 m Ws F P N. 8 c 3. m/s m We then multiply by the time to get the total momentum transferred: p FT 89 N 36 s 6.79 kg m/s. 5