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A) The head of the image will disappear. B) The feet of the image will disappear. C) The image will become dimmer. D) Nothing will happen to the image. d. [5 pts] An electron starts at rest in a uniform electric field. Fill in the blanks: the electron will move towards _____ electric potential, and the work done by the electric field on the electron will be _____. A) higher; positive B) lower; positive C) higher; negative D) lower; negative e. [5 pts] A battery pack consists of three 1.5-volt batteries connected in parallel. The voltage that the pack can supply to an electrical device is: A) 0.5 V B) 1.5 V C) 4.5 V D) Zero, because batteries don’t work if they are connected in parallel. Physical Sciences 2 Problem 1 (continued) Tuesday, April 7, 2009 Page 4 of 18 d. [4 pts] Two identical capacitors are connected in series to a battery as shown. If you insert a dielectric into capacitor 1, the energy stored in capacitor 2 will ... A) decrease B) remain unchanged C) increase D) cannot be determined from the information given Physical Sciences 3 Saturday, May 12, 2012 PROBLEM 1 [10 pts] The following1:two multiple-choice questions not related. Problem Multiple Choice [40arepts] Clearly circle the best answer to each question. Credit will be given for the correct answer with no explanation required and no partial credit. For each of the following questions, circle the letter corresponding to the best answer from given.a No partial credit field will be given, where Youfield do not 1. Inthe theoptions figure below, uniform magnetic points intoexcept the page. (Thenoted. magnetic vectors needaretoindicated show your work, although you are welcome to use the space on the page for charges by the ⊗ symbols). Four particles with the same mass and different electric scratch paper. follow the paths shown as they pass through this magnetic field with identical, constant speeds. Which of the following correctly ranks the charges from least to greatest? (Remember, any a) [5negative pts] One end ofis asmaller string than withany linear massnumber). number positive density µ is attached to a wall. The other end is attached to a mass M and draped over a pulley L away. A standing (a) located q1 < q3 <a qdistance 2 < q4 wave at the fundamental frequency f1 is set up (b)the q4 string < q2 < is q3 plucked < q1 when in the middle. Which the (c) of q4 < q1following < q3 < q2 will increase the value of f1(d) ? Choose all that apply; partial credit q2 < q3 < q1 < q4 will be awarded. (e) q4 < q2 < q1 < q3 (f) q3 < q1 < q2 < q4 A) Replace the string with one that has a larger linear mass density B) Replace the string with one that has a smaller linear mass density C) Move the pulley closer to the wall D) Pluck the string closer to the wall 2. A 300 square the wave is theofinput E) Hz Increase value M to a high-pass RC filter with a 1500 W resistor and a 0.1 µF capacitor. Which of the following best describes the output of the filter, relative to the input? (a) It is smoother-looking, and has almost the same amplitude. b) [5 pts] An electron is traveling towards the right at a (b)ofIt viswhen smoother-looking, and has with a significantly reduced amplitude. speed it enters a chamber a magnetic ! field B . Which direction must the magnetic field point (c) It is more jagged-looking, and has almost the same amplitude. in order for the electron to exit through the top slit? (d) It is more jagged-looking, and has a significantly reduced amplitude. (e) It has the same shape, and has almost the same amplitude. Upthe same shape, and has a significantly E) Up/Rightreduced amplitude. (f) A) It has B) Down F) Down/Left C) Right G) Out of the page D) Left H) Into the page Page 3 of 19 D) f new = f old 2 E) f new = 2 f old For each of the following questions, circle the letter corresponding to the best answer € € € from the options given. No partial credit will be given, except where noted. You do not need to show your work, although you are welcome to use the space on the page for € € scratch paper. c. [4 pts] Which of the following equations are correct a) [5 pts] of Two strings of rules equalfor lengths but different applications Kirchhoff’s the circuit at right?mass densities are knotted together and stretched between two fixed ends. A particular frequency produces the following (Choose all that apply; partial credit will be given.) standing wave on each string. Which string has the greater density (i.e., mass per unit length)? A) I3 = I4 + I5 B) ε 2 − I2 R1 − € Q1 Q2 − +I R =0 C1 C2 4 2 C) ε1 + ε 2 − I5 R3 = 0 € D) I1 = −I2 − I3 € E) I5 R3 − I4 R2 = 0 A) String A € B) String B € C) The densities are equal Page 3 of 10 D) Not enough information to answer this question b) [5 pts] Which of the following are correct logic statements for the output X? A) X = NOT(A OR B) AND C B) X = A OR B OR (NOT C) C) X = A AND B AND (NOT C) D) X = (NOT A) AND (NOT B) AND C Page 3 of 17 b. [4 pts] In a certain region, there is a uniform magnetic field pointing up, and a uniform electric field pointing in an unknown direction. An electron moves to the right in this region with constant velocity. Which direction does the electric field point? Physical Sciences 3 Wednesday, May 20, 2009 A) left B) right Problem C) up 1: Multiple Choice [40 pts] D) down For each of the unrelated questions, circle the letter corresponding to the best E) into thefollowing page answerF)from the options given. No partial credit will be given. You do not need to show out of the page your work, although you are welcome to use any blank space for scratch paper. a. [5 pts] Four charges of equal magnitudes are arranged as shown. The dashed lines all equal length. Whatisislocated the direction of side the electric field at bar the point marked with a c.have [4 pts] A loop of wire on the N of a stationary question mark? magnet, as shown. If the loop is pulled towards the magnet, which of the following statements accurately describes the resulting effect on the bar magnet? A) up B) down A) There will be a net force on the magnet to the right, away from the loop. left will be a net force on the magnet to the left, towards the loop. B)C)There C)D)There right will be no net force on the magnet, but there will be a net torque. D) There will be no net force or net torque on the magnet. E) The electric field is zero b. [5 pts] A small loop of wire is placed near a very long straight wire, as shown. The current in the long straight wire is initially zero, but it gradually increases to a maximum upwardPage current, steady for a 3 of stays 6 while, and then gradually decreases back to zero. Suppose we define I, the current in the loop, to be positive for counterclockwise current. Which of the following could be the correct graph of I vs time? Page 3 of 20 Physical Sciences 3 Tuesday, March 1, 2011 Problem 1: Multiple choice [20 pts] For each of the following unrelated questions, circle the letter corresponding to the best answer from the options given. No partial credit will be given. You do not need to show your work, although you are welcome to use any blank space for scratch paper. a. [5 pts] Marisol and Nadir make some observations with three charged blocks—A, B, and C. They see that block A is attracted to block B, but repelled by block C. What should happen if block B is brought near block C? A) They’ll attract one another. B) They’ll repel one another. C) They’ll feel no attraction or repulsion. D) There’s not enough information to determine what will happen b. [5 pts] An electron is traveling with a constant velocity v = v 0 xˆ , when it enters a region of space with a uniform electric field E = E 0 yˆ (see figure). Which of the following trajectories best describes the electron’s motion in this region? € € Page 2 of 10 Physical Sciences 2 Thursday, March 25, 2010 Problem 1: Multiple choice [20 pts] For each of the following unrelated questions, circle the letter corresponding to the best answer from the options given. No partial credit will be given. You do not need to show your work, although you are welcome to use any blank space for scratch paper. a. [4 pts] Two equal and opposite point charges are arranged as shown. Points A, B, and C lie along the dashed line halfway between the charges. At which of the labeled points would the electric field point directly up (in the +y direction)? A) A B) B C) C D) all of the above E) none of the above b. [4 pts] Which of the following statements about equipotential contour lines is true? A) Contour lines begin on positive charges and end on negative charges. B) Contour lines can cross other contour lines, but only at right angles. C) Contour lines are closer together in regions where the electric field is stronger. D) Contour lines are closer together in regions where the electric potential is higher. c. [4 pts] Two capacitors, each with the same initial charge Q, are connected to each other as shown. C1 is greater than C2. When the switch S is closed, what will happen? A) The charge on C1 will decrease, and the charge on C2 will decrease B) The charge on C1 will decrease, and the charge on C2 will increase C) The charge on C1 will increase, and the charge on C2 will decrease D) The charge on C1 will increase, and the charge on C2 will increase E) The charge on both capacitors will remain the same Page 3 of 11 Physical Sciences 3 Tuesday, February 28, 2012 Problem 1: Multiple Choice (continued) e) [5 pts] Consider the magnitude and direction of the electric field at the point indicated by a question mark (?) below. The numbers indicate equally-spaced marks on a ruler; +q represents a point charge at the location indicated. +q 0 1 ? 2 3 4 5 6 7 8 Circle the letter below for each configuration of charges that would give the same magnitude and direction of the electric field at the point indicated by the question mark (?). Circle all that apply; partial credit will be awarded. –q A) 0 1 2 ? 3 4 5 0 1 2 3 0 1 2 3 +4q D) 0 0 5 6 7 8 4 5 6 7 8 5 6 7 8 ? 1 2 3 +2q E) 8 ? +2q C) 4 7 –q ? B) 6 4 ? 1 2 3 4 –2q 5 Page 5 of 11 6 7 8 Physical Sciences 2 Thursday, February 26, 2009 Problem 1: Multiple Choice [16 pts] For each of the following unrelated questions, circle the letter corresponding to the best answer from the options given. No partial credit will be given. You do not need to show your work, although you are welcome to use any blank space for scratch paper. a. [4 pts] Consider a traveling wave that moves in the +x direction on a stretched string. The wavelength is 50 cm, the frequency is 100 Hz, and the amplitude is 1 cm. Which of the following is the correct equation for the wave? ! ! ! ! ! A) B) C) D) E) D(x,t) = (1 cm)sin((2 rad/m)x " (628 rad/s)t ) D(x,t) = (1 cm)sin((12.6 rad/m)x " (0.0628 rad/s)t ) D(x,t) = (1 cm)sin((2 rad/m)x + (0.0628 rad/s)t ) D(x,t) = (1 cm)sin((12.6 rad/m)x " (628 rad/s)t ) D(x,t) = (1 cm)sin((12.6 rad/m)x + (628 rad/s)t ) b. [4 pts] A pipe is open at both ends, and a standing sound wave is established in it with a frequency 3 times the fundamental frequency of the pipe. Fill in the blanks: Including both ends, the pipe contains ____ and ____ for pressure variations. A) 2 nodes; 3 antinodes B) 3 nodes; 2 antinodes C) 3 nodes; 3 antinodes D) 3 nodes; 4 antinodes E) 4 nodes; 3 antinodes c. [4 pts] A loudspeaker emits a spherical sound wave. At a distance of 5 m from the speaker, the loudness of the sound is measured to be 80 dB. What is the power of the sound emitted by the speaker? (Note: the reference intensity I0 is 10-12 W/m2.) A) 2.5 × 10-7 W B) 3.2 × 10-7 W C) 1.0 × 10-4 W D) 2.5 × 10-3 W E) 3.1 × 10-2 W Page 3 of 6 Physical Sciences 2 Tuesday, April 7, 2009 Problem 1: Multiple Choice [20 pts] For each of the following unrelated questions, circle the letter corresponding to the best answer from the options given. No partial credit will be given. You do not need to show your work, although you are welcome to use any blank space for scratch paper. r a. [4 pts] A dipole moment p points to the right, as shown below. What is the direction of the electric field at point A due to the dipole? (Remember, this is a physics course, not a chemistry course!) ! A) up B) down C) left D) right E) the electric field is zero b. [4 pts] Which of the following equations is a valid statement of the loop rule for the left-hand loop in the circuit shown? Q =0 C Q B) E " I1R + = 0 C Q C) E + I1R " = 0 C Q D) E + I1R + = 0 C E) none of them is valid A) E " I1R " ! ! ! c.![4 pts] An electron initially at rest accelerates to the left under the influence of an electric field. Fill in the blanks: In the vicinity of the electron, the electric field points to the ____, and the electron moves towards _____ electric potential. A) left; lower B) left; higher C) right; lower D) right; higher Page 3 of 11 Physical Sciences 3 Mar. 20, 2008 PROBLEM 1 [20 pts] The following multiple-choice questions are not related. Circle the best answer to each question. Credit will be given for the correct answers with no explanation required and no partial credit. 1. Four electric charges with equal magnitudes are arranged on corners of a square. What is the direction of the resulting electric field at location marked with a ? indicate positive charges, indicate negative charges. (a) Upwards, (b) Downwards, (c) Left, (d) Right, (e) Up / Left, (f) Up / Right, (g) Down / Left, (h) Down / Right, 2. Consider two wires made of the same material. Wire #2 has twice the diameter of Wire #1, and is twice as long as Wire #1. The resistance between the ends of Wire #2 is _________. (a) One fourth the resistance of Wire #1. (b) Half the resistance of Wire #1. (c) The same resistance as Wire #1. (d) Twice the resistance of Wire #1. (e) Four times the resistance of Wire #1. 2 Physical Sciences 3 Mar. 20, 2008 PROBLEM 1 [continued] 3. In the circuit below, charge Q0 and voltage V0 are placed across capacitor C1. Capacitor C2 is initially uncharged, and C2 is much smaller than C1. Fill in the blanks: After a long time the switch S is closed, the charge on C2 will be ____________ and the voltage across C2 will be ____________. (a) much less than Q0; much less than V0 (b) about equal to Q0; much less than V0 (c) much less than Q0; about equal to V0 (d) about equal to Q0; about equal to V0 Short-answer question: 4. A defibrillator must deliver 50 J of energy across the chest of a heart attack victim over a time of roughly 10 ms in order to put the patient’s heart back into normal rhythm. It does this by discharging a capacitor through the chest wall, which has a resistance of 100 !. What approximate capacitance should the defibrillator have, and what voltage should be placed across the capacitor when it is initially charged? Capacitance = __________ Voltage = __________ 3 Physical Sciences 2 Tuesday, February 23, 2010 Problem 1: Multiple choice [15 pts] For each of the following unrelated questions, circle the letter corresponding to the best answer from the options given. No partial credit will be given. You do not need to show your work, although you are welcome to use any blank space for scratch paper. a. [5 pts] In the diagram at right, the dashed curves represent equipotential contour lines around a twodimensional charge distribution. (The charges themselves are not shown.) Which of the arrows on the figure is the best representation of a correct electric field line? A) B) C) D) b. [5 pts] A parallel-plate capacitor is charged until 100 mJ of electrical energy are stored in it. Then it is carefully disconnected from the charging circuit, and the plates are pulled apart so that the separation distance between the plates doubles. Afterwards, how much electrical energy is stored in the capacitor? A) 25 mJ B) 50 mJ C) 100 mJ D) 200 mJ E) 400 mJ c. [5 pts] In a certain region of space, there is a uniform electric field pointing to the right. One at a time, you place an electron, a proton, and a dipole into this region and release them. Which way would each one accelerate? A) electron right, proton left, dipole left B) electron left, proton right, dipole right C) electron left, proton right, dipole left D) electron right, proton left, dipole would not accelerate E) electron left, proton right, dipole would not accelerate Page 3 of 6 Physical Sciences 3 Tuesday, May 11, 2010 Problem 1: Multiple choice [40 pts] For each of the following unrelated questions, circle the letter corresponding to the best answer from the options given. No partial credit will be given. You do not need to show your work, although you are welcome to use any blank space for scratch paper. a. [5 pts] The diagram below shows electric field lines near two point charges of equal magnitude. What can you conclude about the signs of the charges? A) Both are positive. B) Both are negative. C) The charges have the same sign, but it is not possible to tell whether they are both positive or both negative. D) The charges have opposite signs, but it is not possible to tell which is which. E) It is not possible to tell whether the charges have the same or opposite sign. b. [5 pts] A negative point charge is located above a neutral conducting sheet, as shown. In equilibrium, what is the direction of the electric field at point P, just above the conductor? A) up B) down C) up and left D) down and right E) The electric field is zero at P. Page 3 of 18 Physical Sciences 3 Tuesday, February 28, 2012 Problem 1: Multiple Choice [25 pts] For each of the following questions, circle the letter corresponding to the best answer from the options given. You do not need to show your work, although you are welcome to use the space on the page for scratch paper. Partial credit will not be awarded, unless otherwise stated. a) [5 pts] You place an electric dipole in a uniform electric field with field lines as shown. In which configuration would the dipole experience a counterclockwise torque? A) B) C) D) b) [5 pts] You have a tank that contains an aqueous solution of ions. As shown in the figure at right, you connect a battery across two electrodes that cover the areas A at each end of the tank. The electrodes are separated by a distance L. Which of the following changes would increase the amount of current flowing through the tank? Circle all that apply; partial credit will be awarded. A) Increasing the distance L. B) Increasing the voltage ΔV. C) Adding some additional ionic solute to the solution. D) Adding some corn syrup to increase the viscosity (drag) of the solution. E) Increasing the electrode area A. Page 3 of 11 as shown. Which of the five possible paths shown would the light ray follow? A) path a B) path b PhysicalC)Sciences path c 3 Wednesday, May 20, 2009 D) path d Problem 1 (continued) E) path e d. [5 pts] Three point charges (–Q, –Q, and +2Q) are located at the corners of a square, as shown. What is the direction of the electric c. [5 pts] the following statements must be true about an electron moving in an field at theWhich fourthofcorner? electric field? A) up and left ( ) r A) The electron moves in the direction opposite to E . B) up and right ( ) B) The electron moves towards higher electric potential. C) down and left ( ) C) The force on the electron is perpendicular to its velocity. ! D) down and right ( ) D) The force on the electron is independent of its velocity. E) the electric field is zero Page 3 of 20 e. [5 pts] The graph at right shows the intensity of light as a function of position on the screen in a twoslit interference setup. Fill in the blank: At the second-order maximum on either side, the light from one slit travels ______ the light from the other slit. A) the same distance as B) twice as far as C) one wavelength further than D) two wavelengths further than E) three wavelengths further than f. [5 pts] Consider the transistor logic circuit shown below. It has one input (A) and one output (X). Fill in the blanks: When A is true, X is ___; and when A is false, X is ___. A) true; true B) true; false C) false; true D) false; false Page 4 of 20 Physical Sciences 3 Saturday, May 12, 2012 Problem 9: At Capacity [15 pts] Four capacitors, each with the same capacitance C, are connected to a battery as shown. a) [7 pts] When fully charged, how much potential energy is stored in each capacitor? b) [8 pts] What is the change in the total energy stored in the capacitors after a dielectric with κ = 3 is inserted into capacitor C2? Assume that the battery remains connected during the process. Page 15 of 17 Physical Sciences 3 Wednesday, May 20, 2009 Problem 4: Transmission Line [20 pts] A DC transmission line is made up of two parallel wires strung horizontally, separated by a distance d. They transmit electrical power P at a very high voltage ΔV by carrying equal currents in opposite directions. a. [5 pts] What is the magnitude of the magnetic field due to these two currents at the midpoint between the wires? b. [5 pts] Do the wires attract or repel each other? c. [5 pts] What is the magnitude of the force per unit length that the wires exert on each other? d. [5 pts] Calculate, to 1 significant digit, the magnitude of the force between the wires over a 100 meter-long section of the transmission line with ΔV = 110 kV, P = 5 × 107 W, and d = 1 m. Page 8 of 20 Physical Sciences 3 Wednesday, May 20, 2009 Problem 2: How Wide? [10 pts] Two point sources of 500-nm light are separated by a distance of 1 cm. Light from the two sources passes through a narrow slit 10 m away, and then falls on a screen of photographic paper 1 m behind the slit, as in the figure (which is not drawn to scale). What is the minimum width of the slit such that the two images will be resolvable on the photographic paper? Page 6 of 20 Physical Sciences 3 Tuesday, April 12, 2011 Problem 2: You Can’t Handle The Truth….Tables! [16 pts] a) [8 pts] Draw a logic gate diagram to implement the following truth table using some combination of NOT, AND, and OR gates only. Clearly label your inputs and outputs. A 0 0 1 1 B 0 1 0 1 X 1 1 0 1 b) [8 pts] Complete the truth table for the transistor circuit shown below. You do not need to show your work. Note that the input B appears twice in the circuit. A 0 0 1 1 B 0 1 0 1 X Page 5 of 10 Physical Sciences 3 Tuesday, February 28, 2012 Problem 2: I’m Positive [15 pts] You discover a new cell type and measure the relative electrical potential at two points across the cell membrane as indicated on the figures below. a) [6 pts] On the figure below, sketch the charge distribution and the electric field lines that would be consistent with these potential measurements. •V=0 b) [4 pts] On the figure below, sketch the equipotential surfaces that would be consistent with these potential measurements. •V=0 fluid outside fluid outside membrane membrane fluid inside fluid inside • V = +50 mV • V = +50 mV c) [5 pts] A chloride ion (Cl–) is located in an ion channel at the center of the membrane of this cell. If the membrane has a thickness of d = 6 nm, and a dielectric constant of κ = 9, calculate the magnitude and direction of the electrostatic force on this ion. (e = 1.6 × 10–19 C) Page 6 of 11 Physical Sciences 3 Tuesday, May 11, 2010 Problem 8: Resistor network [15 pts] In the circuit shown at right, the battery emf is 12 V, and each of the resistors is 200 Ω. a. [10 pts] Calculate the current through each resistor in the circuit. b. [5 pts] Calculate the power output of the battery. Page 13 of 18 Physical Sciences 3 Saturday, May 12, 2012 Problem 5: Lenses Law [20 pts] A 4-cm bright object is place in front of two thin lenses—the first has a focal length of fA = –2 cm and the second has a focal length fb = 3 cm—as shown below: a) [8 pts] On the diagram above, use the principal rays to show where the final image is formed. Clearly indicate the position of the final image in your diagram. b) [6 pts] What is the (exact numerical) location of the final image on the number line? c) [6 pts] How tall is it? Is it upright or inverted? Is it a real or virtual image? Page 11 of 17 Physical Sciences 3 Tuesday, April 12, 2011 Problem 5: Digital circuits [15 pts] a. [7 pts] Draw a logic diagram to implement the following truth table using some combination of AND, OR, and NOT gates. Clearly label your inputs and output. A B X 0 0 0 0 1 1 1 0 0 1 1 0 b. [8 pts] Complete the truth table below for the transistor circuit shown. You do not need to show your work. A 0 0 1 1 B 0 1 0 1 X Page 8 of 9 Physical Sciences 2 Thursday, March 25, 2010 Problem 6: Carrying charges [15 pts] In the circuit shown at right, the three capacitors all have the same capacitance C. They are initially uncharged. At time t = 0, the switch S is closed. a. [5 pts] Determine the time constant ! for charging the capacitors, in terms of R and C. b. [10 pts] A very long time after the switch is closed, how much energy is stored in each of the capacitors (1, 2, and 3)? Give your answers in terms of E and C. Page 10 of 11 Physical Sciences 3 Saturday, May 12, 2012 Problem 8: One if by Land, Two if by Sea [20 pts] You are the sexton in the Old North Church in Boston. It is April 18, 1775. You need to send a signal to Paul Revere, who is 3 km away. a) [10 pts] You could choose to hang either one or two lanterns in the church tower. If Paul Revere’s pupils have a diameter of 4 mm, and the light from your lanterns has a wavelength of 580 nm, what is the minimum spacing between the lanterns required for him to see them as two distinct lanterns? For simplicity, treat the pupil as a simple circular aperture; you do not need to consider the index of refraction inside the eye. Assume that diffraction is the only limitation on visual acuity. b) [10 pts] Another possibility would be to ring the church bell either once or twice. For Paul Revere to hear the bell clearly, it should have a loudness of at least 40 dB where he is standing, 3 km away. What would be the loudness of the bell for you, the sexton, standing only 1 meter away? Assume the sound is distributed evenly in all directions. Page 18 of 19 Physical Sciences 2 Tuesday, April 7, 2009 Problem 5: Digital Circuits [15 pts] a. [7 pts] Draw a logic diagram to implement the following truth table using some combination of AND, OR, and NOT gates. Clearly label your inputs and output. A B X 0 0 1 0 1 0 1 0 0 1 1 1 b. [8 pts] Complete the truth table below for the transistor circuit shown. You do not need to show your work. A 0 0 1 1 B 0 1 0 1 X Page 9 of 11 Physical Sciences 3 Mar. 20, 2008 PROBLEM 9 [10 pts] Coherent light of frequency 9 × 1014 Hz passes through two thin slits and falls on a screen 81 cm away. You observe that the third bright fringe occurs at ±3 cm on either side of the central bright spot. (a) How far apart are the slits? (b) At what distance from the central bright fringe will the third dark fringe appear? Physical Sciences 2 Tuesday, April 7, 2009 Problem 6: Two-loop Circuit [15 pts] a. [8 pts] For the circuit shown below, clearly label the currents in the circuit. Using Kirchoff’s rules, write down a set of equations that you could use to solve for the currents, clearly indicating where each equation comes from. You do not need to solve your equations. b. [7 pts] You remove battery !1 and replace it with a wire, as shown below. Find an expression for the current through resistor R5 in the new circuit. Page 9 of 10 Physical Sciences 3 Saturday, May 12, 2012 Problem 4: Hear Ye! [15 pts] The ear canal (external auditory canal) can be treated as a tube that is open at one end and closed at the other, with a length of L = 2 cm. a) [10 pts] On the tubes shown below, sketch the standing waves formed by the pressure variations ΔP for the three lowest resonant frequencies of the ear canal. Then calculate these frequencies; the speed of sound in air is v = 340 m/s. b) [5 pts] Write an equation that describes the pressure variation ΔP as a function of x and t inside the ear canal, for the lowest resonant frequency, in terms of its amplitude ΔP0. Take x = 0 to be the outer (open) end of the ear canal for this standing wave. Be sure to give numerical values for all parameters in your equation (except for ΔP0). Page 9 of 19 Physical Sciences 3 Wednesday, May 20, 2009 Problem 4: Digital Circuits [15 pts] a. [7 pts] Draw a logic diagram to implement the following truth table using some combination of AND, OR, and NOT gates. Clearly label your inputs and output. A B X 0 0 0 0 1 1 1 0 0 1 1 0 b. [8 pts] Complete the truth table for the transistor circuit shown below. You do not need to show your work. A 0 0 1 1 B 0 1 0 1 X Page 9 of 20 Physical Sciences 3 Tuesday, May 11, 2010 Problem 6: Film session [15 pts] You would like to coat a piece of glass (n = 1.4) with a very thin coat of transparent film (n = 1.3) so that when the glass is viewed at normal incidence, it strongly reflects blue light of wavelength 480 nm, but does not reflect any red light of wavelength 640 nm. How thick should the film be? (There are many possible correct answers; you only need to calculate one.) Page 11 of 18 Physical Sciences 3 Tuesday, May 11, 2010 Problem 8: Diffraction [10 pts] Light of wavelength 550 nm passes through a tiny pinhole of diameter D = 50 microns, as in the figure (not to scale). What will be the radius of the central Airy disc on a screen located 1 meter behind the pinhole? Page 14 of 20 Physical Sciences 3 Mar. 20, 2008 PROBLEM 4 [10 pts] (a) In the circuit below, the battery has !V = 9 volts and the resistors have R1 = R2 = R3 = R4 = 100 ". Calculate the current through the resistor R4. I4 = _______________ (b) For the circuit below, label the currents in the circuit using arrows. Use Kirchhoff’s rules (junction and loop rule) to write down the equations you would need to solve in order to find all the currents in the circuit (assume you know the resistor values and battery voltages). You do not need to solve these equations. 7 Physical Sciences 3 Wednesday, May 20, 2009 Problem 8: Fluorescent Bead [15 pts] A bead of diameter dbead = 1 µm is coated with a fluorescent substance that absorbs green light of wavelength λex = 520 nm and emits red light of wavelength λem = 640 nm. The bead is placed in the path of a 520-nm laser. The laser emits with power P = 1 mW; assume that the laser beam has a diameter of dlaser = 1 mm and that the intensity is uniform within the beam. (Planck’s constant h = 6.63 × 10–34 J⋅s. a. [8 pts] Assume for the sake of simplicity that every 520-nm photon that strikes the bead causes a fluorescence. How many red photons does the bead emit per second? b. [7 pts] When the bead fluoresces, it emits red light isotropically (with equal intensity in all directions). A photodetector calibrated to detect 620-nm light is placed at a distance L = 10 cm from the bead. What is the intensity of red light incident on the detector? Page 14 of 20 Physical Sciences 2 Tuesday, April 7, 2009 Problem 5: Digital Circuits [15 pts] a. [8 pts] Draw a logic diagram to implement the following truth table using some combination of AND, OR, and NOT gates. Clearly label your inputs and output. A B X 0 0 0 0 1 0 1 0 1 1 1 0 b. [7 pts] Complete the truth table for the transistor circuit shown below. What is the common name of this gate? Page 8 of 10 Physical Sciences 3 Mar. 20, 2008 PROBLEM 7 [12 pts] A string lies along the x-axis and supports vibrations in the y-direction. Such transverse waves have a wave speed of 8 m/s, amplitude of 0.07 m, and a wavelength of 0.32 m. The waves travel in the –x direction, and at time t = 0 the point x = 0 has its maximum upward displacement, i.e. D(x=0,t=0) = 0.07 m. (a) Find the frequency and period of these waves. (b) Write the function describing the displacement of the string at all locations and times. (c) Find the transverse displacement at the location x = 0.36 m at time t = 0.175 s. (d) How much time must elapse from the instant in part (c) until the piece of the string at the point x = 0.36 m next has maximum upward displacement? Physical Sciences 3 Mar. 20, 2008 PROBLEM 5 [20 pts] The capability a small camera has to generate very bright flashes of light is made possible by using a capacitor to store energy slowly and release it quickly. Consider the following circuit representing the flash component of a camera: (a) Initially, the switch S is set to connect the capacitor to the battery. What is the time constant for charging the capacitor? !charge = ___________ (b) How much energy is stored in the capacitor? Ustored = ____________ (c) When a picture is taken, the switch S moves to connect the capacitor to the flash bulb, which acts as a very small resistor. How much time does it take for the capacitor to lose 63% of its charge? t = ______________ (d) Estimate the value of the power dissipated in the bulb during the discharge. P " _____________ 8 Physical Sciences 3 Saturday, May 12, 2012 Problem 3: Doppler Difference [15 pts] The Doppler Effect for sound is different depending on whether the source is moving or the observer is moving. Let’s explore some consequences of this difference. a) [7 pts] In one type of Doppler effect, the apparent frequency can become effectively infinite. Which case is this (source moving or observer moving)? Under what conditions will the frequency become infinite? Support your answer with an explicit formula. b) [3 pts] If the frequency is effectively infinite, as calculated in part (a) above, what does that mean about the wavelength? c) [5 pts] Which of the following diagrams best represents the wavefronts for the Doppler effect in the case of an infinite frequency? (circle one) A) B) C) Page 8 of 19 D) Physical Sciences 3 Wednesday, May 20, 2009 Problem 10: Capacitors [20 pts] Two capacitors, C1 = 1 µF and C2 = 2 µF, are connected in series to a 6-volt battery via a resistor of R = 10 kΩ, as shown. Both capacitors are originally uncharged. a. [5 pts] Calculate the final charge on each capacitor. b. [4 pts] How much time (after the battery is connected) elapses until capacitor 1 reaches 90% of its final charge? Page 15 of 20 Physical Sciences 3 Wednesday, May 20, 2009 Problem 11: Falling Magnet [10 pts] A circular loop of wire lies flat on a table. A permanent magnet is held over the center of the loop with its N pole pointing down, and then released. a. [3 pts] As the magnet starts to fall, what would be the direction of the magnetic field generated by the induced current? b. [3 pts] Would the induced current flow in the loop clockwise or counterclockwise (as viewed from above)? c. [4 pts] Would the current in the loop exert an upward, downward, or zero net force on the magnet? Explain. Page 17 of 20 Physical Sciences 2 Tuesday, April 7, 2009 Problem 3: Mystery RC [20 pts] You have been given a black box containing a resistor Rx and capacitor C in series. To determine the values of Rx and C, you set up the circuit shown in the figure at right, and then use Logger Pro to measure the time constant of the circuit. When the value of the known external resistor R1 is 15 kΩ, you measure the time constant to be τ1 = 70 ms. When you replace R1 with a different known resistor R2 = 27 kΩ, the new time constant is τ2 = 82 ms. a. [10 pts] Calculate the mystery resistance Rx and the mystery capacitance C. Page 6 of 11 Physical Sciences 3 Tuesday, May 11, 2010 Problem 9: Hoop it up [15 pts] Consider a thin charged hoop of radius R and charge per unit length + ! uniformly distributed along its circumference. The hoop spins in place with angular velocity ! counterclockwise, as shown. a. [5 pts] The spinning charge constitutes an electric current. What is the magnitude of this current? Express your answer in terms of R, ! , and ! . b. [3 pts] What is the direction of the magnetic field at the hoop’s center? c. [7 pts] Now consider the hoop spinning as shown, with a permanent magnet aligned on the rotational axis of the hoop, with the S pole facing the hoop. Do the hoop and magnet attract each other, repel each other, or neither? Explain. Page 14 of 18 Physical Sciences 3 Wednesday, May 20, 2009 Problem 8: Circuits [15 pts] a. [5 pts] Consider the circuit shown at right. Clearly label the currents with their directions, and then use Kirchhoff’s rules to write down the equations you would need to solve for the currents. b. [5 pts] For R1 = 1 kΩ, R2 = 2 kΩ, R3 = 4 kΩ, and ΔV = 9 V, calculate the values of the currents you labeled in part a. c. [5 pts] How much total power is dissipated in the circuit? Page 12 of 20 Physical Sciences 3 Saturday, May 12, 2012 Problem 7: Silent Killer [15 pts] During a surgical procedure, a thin, steel scalpel blade was left inside a patient’s torso. The doctors are attempting to use ultrasound imaging to find the location of the blade. Sound waves with a frequency of 5 MHz are emitted from the transducer, but the doctor cannot locate the blade. Determine the thickness of the steel blade. (The speed of sound in soft tissue is 1540 m/s; the speed of sound in steel is 6000 m/s.) Page 13 of 17 Physical Sciences 2 Thursday, March 25, 2010 Problem 5: Digital circuits [15 pts] a. [7 pts] Draw a logic diagram to implement the following truth table using some combination of AND, OR, and NOT gates. Clearly label your inputs (A and B) and output (X). A B X 0 0 0 0 1 1 1 0 0 1 1 0 b. [8 pts] Complete the truth table below for the transistor circuit shown. You do not need to show your work. A 0 0 1 1 B 0 1 0 1 X Page 9 of 11 Physical Sciences 3 Tuesday, May 11, 2010 Problem 11: Beam me up, Scotty [15 pts] A laser beam cannot be perfectly collimated; even if the beam is a pure plane wave inside the laser cavity, it will spread out slightly after coming out of the opening due to diffraction. Consider a laser pointer that emits 633-nm red light through an opening of diameter 1.0 mm. What will be the radius of the spot that the beam makes on the ground if you shine it from the top of a building 100 m high? Page 16 of 18 Physical Sciences 3 Saturday, May 12, 2012 Problem 4: Charged Particles [15 pts] a) [7 pts] Two charged particles, q1 and q2, are fixed at the origin and at x = d on the x-axis, respectively. A third charged particle, q3, which is free to move, is in static equilibrium at x = 2d. Find q1 in terms of q2. b) [8 pts] Charge q3 is now placed at the location where the electric potential due to the fixed charges is zero. Find this location x in terms of d. Assume the charges are the same as part a) and that V = 0 at infinity. (You cannot place the charge at infinity.) Page 10 of 17 Physical Sciences 2 Thursday, April 23, 2009 Problem 2: Mirror, Mirror [15 pts] You look into a curved mirror at a distance of 30 cm. The image of your face in the mirror is upright and half the size of your actual face. a. [5 pts] What is the image distance? Is the image real or virtual? b. [5 pts] What is the radius of curvature of the mirror? Is the mirror concave or convex? c. [5 pts] Draw a ray diagram for the image of your face in the mirror. Your diagram should include the object, image, focal point, center of curvature, and three principal rays. It does not need to be exactly to scale. Page 5 of 6 Physical Sciences 2 Tuesday, February 23, 2010 Problem 2: Looking for zero [15 pts] A point charge Q is located at the origin, and another charge, q, is located at the point (a, 0) of the coordinate system shown in the figure. a. [8 pts] If the electric field at the point (–a/2, 0) is zero, what is q in terms of Q? b. [7 pts] If, instead, the electric potential at the point (–a/2, 0) is zero, what is q in terms of Q? Assume V=0 at infinity. Page 4 of 6 Physical Sciences 3 Saturday, May 12, 2012 Problem 2: Near and Far [15 pts] Your friend Qiu has an unaided near point of 10 cm and an unaided far point of 50 cm. a) [4 pts] When she focuses on an object at her near point, what is the effective focal length of her corneal lens? The retina is 2.5 cm behind the lens; treat the corneal lens as a single “thin lens” and ignore the fluid inside the eyeball. b) [6 pts] Qiu stands 5 cm away from a plane mirror and looks directly at her own eye. She can see her eye clearly in the mirror. Construct a ray diagram below that shows how an image of her eye (specifically, her cornea) is formed on her retina. (Note that the object in question is located at her corneal lens—she’s looking at her own cornea.) F retina F corneal lens mirror c) [5 pts] Qiu’s optometrist prescribes eyeglasses that sit 2 cm away from her eyes. What is the focal length of her prescription lenses? Are they converging or diverging? Page 7 of 19 Physical Sciences 3 Tuesday, April 12, 2011 Problem 4: Kirchoff’s Revenge [15 pts] Consider the following circuit: a) [10 pts.] If ε1 = 6 V and ε2 = 12 V, find the current through the 2-Ω resistor. b) [5 pts.] How much power is dissipated as heat through the 2-Ω resistor? Page 7 of 9 Physical Sciences 2 Tuesday, April 20, 2010 Problem 2: Current events [15 pts] A square loop of wire sits to the left of a long straight wire as shown. The straight wire carries a current that is upward and increasing in time. a. [5 pts] Is the induced current in the loop clockwise, counterclockwise, or zero? Explain. b. [10 pts] In which direction is the net magnetic force on the loop? Explain. Page 5 of 6 Physical Sciences 3 Tuesday, May 11, 2010 Problem 10: Sounding off [20 pts] A standing sound wave in air has the equation P ( x,t ) = Patm + !P sin ((15.7 rad/m ) x ) cos (" t ) where !P and ! are constants. The wave sounds in a tube of length 60 cm, which is open at both ends (x = 0 and 60 cm). Use v = 340 m/s for the speed of sound. a. [8 pts] At what values of x are there antinodes for pressure variation? b. [4 pts] Calculate the constant ! . c. [8 pts] What would be the fundamental frequency of sound in this tube? Page 15 of 18 Physical Sciences 3 Tuesday, February 28, 2012 Problem 4: Charge It [20 pts] A certain cell has a membrane with a total surface area A = 0.01 mm2, a thickness d = 6 nm, and a dielectric constant κ = 9. a) [8 pts] If the potential of the cell interior (with respect to the exterior) is V = –70 mV, find the charge on the inside of the membrane, and the charge on the outside of the membrane. b) [7 pts] Suppose the cell membrane has become discharged (so there is no potential difference across the membrane). Calculate the minimum number of molecules of ATP that would be required to do the electrical work of re-charging the membrane back to ΔV = –70 mV. (Hydrolysis of one molecule of ATP yields approximately 7 × 10–20 J.) continued… Page 8 of 11 Physical Sciences 3 Tuesday, February 28, 2012 Problem 4 continued c) [5 pts] The cell potential is maintained in part by an ion pump known as the Na/K ATPase, which pumps 3 Na+ ions out and brings 2 K+ ions in to the cell with each cycle. The net electrical result is the transfer of a single elementary charge (1.6 × 10–19 C) from the interior to the exterior of the cell with each cycle. When the membrane is fully charged at ΔV = –70 mV, how much electrical work is done by one cycle of the Na/K ATPase? Page 9 of 11 Physical Sciences 3 Wednesday, May 20, 2009 Problem 12: Antireflective Coating [10 pts] Lenses in high-precision optical equipment are sometimes covered with an antireflective coating so as to eliminate reflected light at a particular frequency (and nearly eliminate it at nearby frequencies). Suppose you have a piece of glass (nglass = 1.50) coated with a layer of antireflective material t = 110 nm thick. You observe that at normal incidence, green light of wavelength λ = 550 nm is not reflected at all from the surface. a. [5 pts] What is the index of refraction of the antireflective material? (You may assume that it is between 1 and nglass.) b. [5 pts] Are there any wavelengths in the visible spectrum that would be strongly reflected at normal incidence? If so, what are they? Page 18 of 20 Physical Sciences 3 Tuesday, May 11, 2010 Problem 4: Digital/binary [15 pts] a. [7 pts] Draw a logic diagram to implement the following truth table using some combination of AND, OR, and NOT gates. Clearly label your inputs and output. A B X 0 0 1 0 1 1 1 0 0 1 1 1 b. [8 pts] Add the binary numbers 11011 and 1100, and then convert the answer back into decimal. Page 8 of 18 Physical Sciences 3 Wednesday, May 20, 2009 Problem 11: Fiber Optics [10 pts] A fiber optic cable consists of a glass cylinder with index of refraction n1 = 1.50 coated around its sides by a thin layer of cladding with index of refraction n2 = 1.45. Suppose you direct a beam of light from air (nair = 1) into the end of the cable at an angle θ relative to the normal, as in the diagram below. What is the maximum value of θ for which the beam will propagate down the cable undiminished in intensity? (You may neglect absorption.) Page 17 of 20 Physical Sciences 3 Saturday, May 12, 2012 Problem 3: Blowing a Fuse [20 pts] A fuse is a circuit element that allows current to pass freely (like a closed switch) up to a certain limit, but if the current limit is exceeded, the fuse will “blow” and prevent any current from flowing through it (like an open switch). Consider the circuit shown, with a DC power supply of ε = 120 V and a fuse with a limit of 1 amp. R1 (the resistor on the left) = 50 Ω, and C = 1 µF. a) [7 pts] What is the minimum value of R2 (the resistor on the right) for which the fuse will never blow? b) [5 pts] For the value of R2 that you calculated in part a), what will be the final charge on the capacitor? Page 8 of 17 Physical Sciences 3 Saturday, May 12, 2012 Problem 3 (continued) NOTE: For parts c) and d), suppose that R2 = 25 Ω. c) [4 pts] What will be the charge on the capacitor at the instant that the fuse blows? d) [4 pts] What will be the final charge on the capacitor? Page 9 of 17 Physical Sciences 2 Tuesday, April 7, 2009 Problem 2: Shake It [15 pts] You shake a thin wire up and down so that the vertical displacement of any small section of the wire at position x and time t is given by the equation D(x,t) = (3.5 cm)sin[(1.05 rad/cm)x + (35 rad/s)t ] . a. [5 pts] What is the speed of the wave? ! b. [10 pts] Sketch the shape of the wire at time t = 0 in the graph below. Indicate with an arrow the direction in which the wave propagates. Page 5 of 10