Physics 1212 Exam #4A (Final) Instructions: This is a closed-book, closed-notes exam. You are allowed to use a clean print-out of your formula sheet, any scientific calculator, and a ruler. Do not write on your formula sheet, except for your name: it must be handed in, signed but clean, with your exam. There is space after each question to show your work; if you need more space, you may use the back of the page, or request more paper. Please clearly indicate where your work for each problem is. Underline or draw a box around your final answer. The exam consists of six sections. Read all the questions at the start so that you can allocate your time wisely. Do easy ones first! You may not share your calculator. The use of cell phones or any other electronic devices (besides calculators) is prohibited. All such gadgets must be turned off and put away throughout the exam. Do not open the exam until told to begin. You have the one entire class period to finish the exam. Put your last name on every page of the exam and on the formula sheet. You must provide explanations and/or show work legibly to receive full credit for Sections II and III. Make sure that your answers include appropriate units and significant digits. (Note: For intermediate steps in your calculation, it s best to carry more significant digits.) Fundamental constants and unit prefixes are on the Formula Sheet, last page. By signing below, you indicate that you understand the instructions for this exam and agree to abide by them. You also certify that you will personally uphold the university s standards of academic honesty for this exam, and will not tolerate any violations of these standards by others. Unsigned exams will not be graded. Signature: UGACard #: Copyright c 2015 University of Georgia. Unauthorized duplication or distribution prohibited.
Section I II III IV V VI Score /60 /35 /35 /35 /35 /10 I: Multiple-Choice Questions (60 points) For each question below, choose the single best response and write the corresponding capital letter in the box provided. There is no penalty for guessing the wrong answer. 1. A real object is placed in front of a converging lens, at a distance smaller than the focal length. When the object is moved slightly away from the lens, without crossing the focal point, in which direction does the image move or change its size? A. The image moves farther away from the lens. B. The image (lateral) size shrinks. C. The image moves closer to the lens. D. The image position doesn t change. E. Not enough information is given. 2. Two identically shaped solid blocks, S and T, made from two different transparent materials, are immersed in the same liquid L. A ray of light strikes each block at the same angle of incidence, as shown. According to the figure below, what is the relative magnitude of the indices of refraction of the solid blocks, n S and n T, and liquid, n L? L L S T A. n S < n T < n L ; B. n T < n L < n S ; C. n T < n S < n L ; D. n S < n L < n T. E. n L < n S < n T ; Copyright c 2015 University of Georgia. 2
3. Sound waves (including ultrasound) have a speed of wave propagation v Air = 346m/s in air and v Water = 1497m/s in water. Also, note that sin(13.364 o ) = 346/1497. A narrow ultrasound beam striking the flat water surface of your swimming pool A. will have an angle of refraction greater than the angle of incidence if the beam is incident from below the water surface. B. will undergo total internal reflection if incident from below the water surface with an angle of incidence of 8.5 o ; C. will undergo total internal reflection if incident from above the water surface with an angle of incidence of 8.5 o ; D. will undergo total internal reflection if incident from below the water surface for any angle of incidence greater than 13.364 o ; E. will have an angle of refraction smaller than the angle of incidence if the beam is incident from below the water surface; 4. A diffaction grating (in air or vacuum), is illuminated by coherent (laser) light with wavelength λ and wave oscillation period τ = λ/c. The 4th order intensity maximum, to the right of the central intensity maximum M, is located at point Q, as shown in the figure below. A wave crest A, from slit R, and a wave crest B, from the neighboring slit S to the right of R, have departed at the same time from their respective slits of origin. Therefore, at Q, M Q Fig. 2.29 R S Diffr. Grating Laser Beam A. A and B will arrive at the same time. B. B will arrive 3 periods before A. C. B will arrive 4 periods before A. D. A will arrive 3 periods before B. E. A will arrive 4 periods before B. Copyright c 2015 University of Georgia. 3
5. In the figure below, Q 1 is a positive and Q 2 a negative point charge and Q 1 and Q 2 are of comparable magnitude. The point P and the locations of Q 1 and Q 2 form the three corners of a square. Which arrow drawn at P could correctly represent the electric field vector E generated by Q 1 and Q 2 at P? (D) P Fig. 2.16 (E) (B) (C) (A) Q 1 Q 2 A. B. C. D. E. 6. A positron is a sub-atomic particle with the same mass, but the opposite charge as an electron. If an positron is initially placed at a finite distance d from a negative point charge, Q, the positron must be given a certain minimum initial speed, v min, in order to be able to travel away, to a very large (infinite) distance from the point charge Q. What would be the value of this minimum required speed, v min, if instead the positron were placed at a distance d = 8d from another negative point charge, Q = 6Q? A. v min = 6 8 v min B. v min = 4 3 v min C. v min = 3 8 v min D. v min = 3 2 v min E. v min = 6 8 v min Copyright c 2015 University of Georgia. 4
7. In an electric generator, a single, square-shaped metallic wire loop, of 7cm sidelength, is spinning in a uniform magnetic field B at 2100RPM which produces a maximum induced electromotoric force (EMF) of 12mV in the loop. What is the maximum induced EMF in a loop of 14cm sidelength, spinning at 700RPM in the same B-field? A. 2mV B. 72mV C. 18mV D. 8mV E. 16mV 8. A very thin circular ring with radius R = 5cm is centered at the coordinate origin O (0, 0, 0), as shown in Fig. 3.34. The ring lies in the x z-plane with current I 1 flowing around the ring in the direction indicated in panel (B) of Fig. 3.34. (A) (B) Fig. 3.34 y z I 2 I 1 I 2 x x y z z x y x x-y-plane View x-z-plane View An infinitely long straight wire carries a current I 2 = π 40A parallel to the z-axis and crosses the x-axis at x = +8cm, as also shown in Fig. 3.34. The net magnetic field, B, generated by I 1 and I 2, is zero at the center of the ring. What is the magnitude of I 1 and the direction of I 2? A. I 1 = π 50A, I 2 in z-direction; B. I 1 = 50A, I 2 in z-direction; C. I 1 = 25A, I 2 in +z-direction; D. I 1 = 50A, I 2 in +z-direction; E. I 1 = 25A, I 2 in z-direction; Copyright c 2015 University of Georgia. 5
II: A Candle in the Mirror (35 points) A burning candle, 12cm tall, is placed 20cm from a spherical mirror which produces an erect image of the candle, 18cm tall. It is critically important that you make drawings of the positioning of the various elements (optical axis, mirror, candle, focal point, mirror s center, screen), as needed, in each of the three parts of this problem (a) What is the mirror s focal length and what is the radius of the mirror s spherical surface? Is this a convergent (=concave) or a divergent (=convex) mirror? Is the candle s image virtual or real? Hint: Find d from d and m, then f from d and d. (b) At what distance from the mirror would the candle s position on the optical axis coincide with that of its image? A candle s flame is always pointing upward. Would the candle s image at that location show a flame pointing upward or downward? Is the image real or vitual? (c) A projection screen is now placed at some distance from the same mirror and the same candle is placed somewhere between the mirror and the screen. The mirror projects a sharp image of the candle onto the screen and this image is inverted and 36cm tall in absolute height. How far from the mirror is the screen and how far from the screen is the candle? Hint: Get d and d from m and f. Copyright c 2015 University of Georgia. 6
Work and Drawing Space for Problem II: Copyright c 2015 University of Georgia. 7
III: A Capacitor, a Point Charge, and then a Proton (35 points) A planar capacitor of unknown size, location and orientation, stores unknown charges +Q C and Q C, respectively, on its two opposing parallel capacitor plates. It produces an electric field E C [E C,x, E C,y, E C,z ] with components E C,x = 15N/C, E C,y = 25N/C, E C,z = 0, at an observation point, P (x P, 0, 0) with x P =+4m, on the x-axis, both P and the origin O (0, 0, 0) being enclosed between the two capacitor plates. A point charge, Q o = +60nC, is now added, placed at the origin, O, and the capacitor and the point charge combined, then jointly produce a net electric field, E = [E x, E y, E z ]. Assume that the elecric field, EC, produced by the capacitor in the presence of Q o is the same as in the absence of Q o. (a) Calculate the components of E at P ; its strength, E ; and its angle, θ, measured from the +x-direction, with θ > 0 if E points above the x-axis, else θ < 0. On the blank page attached, make a big drawing showing: x-axis; y-axis; O; P ; E C ; the field contribution E o, produced by Q o at P ; and E, as the resultant in a vector addition parallelogram, all field vectors with their tail ends attached to P. (b) Assume each capacitor plate has an area of A=6000m 2 and the space between the two plates is vacuum. What is the absolute amount of charge, Q C, stored on either plate? (c) Now the capacitor is removed and a proton is placed, at rest, at point P and then released, to accelerate subject to the electric field produced by Q o. What is the proton s speed if it travels infinitely far away from Q o? Copyright c 2015 University of Georgia. 8
Work and Drawing Space for Problem III: Copyright c 2015 University of Georgia. 9
IV: Circuit Analysis (35 points) In the circuit shown below, assume that I 2 =8A, I 3 = 3A, R 1 = 2Ω, R 2 = 3Ω, R 4 = 6Ω, R C = 6Ω, and C = 35 µf. Also assume that the capacitor is fully charged, by the voltage drop across it, and there is no current flowing to Fig. 3.66 or from either capacitor plate. a p E I 2 R 1 R 2 x I C e C f y I 3 R 3 R C R 4 b q (a) Find the current, I C. Hint: Use a junction (node). (b) Find the battery voltage E and the resistance R 3. another loop to find R 3. Hint: Use a loop to find E, then (c) Find the total amount of electric field energy stored in the capacitor C. Copyright c 2015 University of Georgia. 10
Work Space for Problem IV: Copyright c 2015 University of Georgia. 11
V: Flux and Induced Current in a Loop (35 points) The infinitely long straight wire in the x-y-plane, shown below at d = 12cm from the x-axis, carries a constant current, I. This currrent produces a magnetic field, B, at the origin, O (0, 0, 0), with a strength of 2.5mT, pointing in the ( z)-direction. A narrow rectangular wire loop in the x-y-plane, with sidelengths a = 24cm and b = 0.4cm, is centered at the origin O (0, 0, 0). Straight Wire y I Fig. 4.17 b d Loop x y x z a (a) Find the current, I, flowing in the straight wire. Is this current flowing in the (+x)- or in the ( x)-direction? (b) Calculate the magnetic flux, Φ m, through the loop, with the loop s area vector, A, defined to point out of the plane of the drawing above. Use the given magnetic field vector, B, at the center of the loop and assume B is approximately uniform across the loop area. (c) Suppose the loop is now made to spin, i.e., rotate, around the x-axis, at an angular speed of ω = 1500radians/s. Find the maximum current value, I max ( current amplitude), of the oscillating current induced in the loop by the magnetic field, B, assuming the loop wire has a resistance of 0.4Ω and, again, that B is approximately uniform across the loop area. Hint: Think of the loop as a coil of 1 turn in an electrical generator. Copyright c 2015 University of Georgia. 12
Work and Drawing Space for Problem V: Copyright c 2015 University of Georgia. 13
VI: Draw a Ray Diagram (Extra Credit: 10 points) A real object is positioned to the right of a convergent mirror, at a distance from the mirror which is less than the absolute value of the mirror s focal length. Use a ruler to draw a clean ray diagram for the formation of the image, showing at least two of the principal rays, the mirror and its focal point, the object and the image, all of them clearly labeled. Also clearly label the incoming and the outgoing side of the mirror. Is the image real or virtual? Warning: If you use the C-ray as one of the principal rays, be aware that it does not pass straight through the mirror, but rather gets reflected at the mirror, subject to the law of reflection: angle of incidence = angle of reflection, with the optical axis being the normal to the mirror surface. Copyright c 2015 University of Georgia. 14