Final Exam: Physics Spring, 2017 May 8, 2017 Version 01

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1 Final Exam: Physics Spring, 2017 May 8, 2017 Version 01 NAME (Please Print) Your exam should have 11 pages. This exam consists of 18 multiple-choice questions (2 points each, worth 36 points), and five longer questions (worth 30 points), for a total of 66 points. By handing in this exam, you agree to the following statement: On my honor, I have neither given nor received unauthorized assistance on this work. Signature

2 1. (2 points) A uniform circular ring of charge Q and radius R is located in the x-y plane, centered on the origin as shown in the figure. 4. (2 points) This circuit has four identical light bulbs (labeled A, B, C, and D). What is the magnitude of the electric field, E, at point P located a distance z above the center of the ring? V A B D (a) kq/r 2 (b) kq/(r 2 + z 2 ) (c) kqz/(r 2 + z 2 ) (d) kqz/(r 2 + z 2 ) (3/2) (e) kqr/(r 2 + z 2 ) (3/2) 2. (2 points) For the previous problem, what is the voltage at location P (assume the voltage is zero at infinity)? (a) kq/r (b) kq/ R 2 + z 2 (c) kq/(r 2 + z 2 ) (d) kqz/(r 2 + z 2 ) (e) kqr/(r 2 + z 2 ) 3. (2 points) A positive charge is inside a conducting spherical shell. The positive charge is not in the center of the shell. Which of the following figures best represents the charge distribution on the inner and outer walls of the shell? C Rank the 4 (A,B,C,D) bulbs from brightest to dimmest (a) A = B > C > D (b) C = D > A = B (c) C > A = B = D (d) C > D > A = B (e) None of these 5. (2 points) For the circuit in the previous question, what is the voltage difference across bulb C? (a) V/3 (b) 2V/3 (c) 3V/5 (d) 2V/5

3 Two resistors are made out of the same material with resistivity ρ. Resistor #2 is twice as long and has a radius twice that of resistor # z direction (out of the page) and a nonzero uniform electric field in the +y direction. Which of the following is a possible trajectory for the particle (purely in the xy plane)? 6. (2 points) If the resistance of resistor #2 is R, the resistance of resistor #1 is (a) 2R. (b) 4R. (c) R/2. (d) R/4. (e) R. 7. (2 points) A positively charged, infinitelylong cylinder of radius R has a uniform charge density ρ (charge per volume). R r L A student wishes to compute the magnitude E of the electric field at a distance r (r < R) from the from the cylindrical axis (that runs along the center). The student writes down Gauss s Law and sketches the centered, cylindrical Gaussian surface S shown. What is the correct expression for the electric flux Φ through this Gaussian surface? (a) Φ = 2πr 2 E (b) Φ = 4πr 2 E (c) Φ = 2πrLE (d) Φ = πr 2 LE (e) Φ = 2πRLE 8. (2 points) A positively charged particle is moving in the xy-plane in a region where there is a non-zero uniform magnetic field B in the 9. (2 points) An AC voltage source drives the circuit with an angular frequency w. 3 (t) C For which of the following frequencies will the current flowing through the resistor be the highest? (a) very low frequencies (b) very high frequencies (c) the frequency w = 1/ LC. L R

4 10. (2 points) A thin wire is shaped into a semicircle of radius R, and has a constant current I flowing through it in a clockwise orientation, as shown below. The semicircle is in a uniform magnetic field, which is coming out of the page. I 12. (2 points) The primary coil of a transformer is hooked up to a 60 Hz AC voltage source, V 1 (t). The secondary coil of the transformer is attached to a resistor with resistance R. I B R What is the magnitude of the net force on the wire due to the magnetic field (ignore the force on the wire segments to the left and right of the semicircle)? (a) F = IRB (b) F = 2IRB (c) F = πirb (d) F = (2 points) A wire loop is being rotated within a uniform magnetic field.. The angular velocity of the loop is a constant value of ω (radians/second). If the surface area of the loop is A, what is the magnitude of the instantaneous induced emf in the loop at the moment as shown in the figure (where the magnetic field lies in the plane of the wire loop)? (a) zero (b) ABω (c) ABω/(2π) (d) 2πABω (e) AB If the resistance R is lowered to R/2, (a) the peak current flowing through the resistor is reduced by a factor of two and the peak current I 1 doubles. (b) the peak current flowing through the resistor doubles and the peak current I 1 is reduced by a factor of two. (c) the peak current flowing through the resistor doubles and the peak current I 1 remains the same. (d) the peak current flowing through the resistor doubles and the peak current I 1 doubles. 13. (2 points) For the transformer in the previous question, if the resistance is lowered to R/2, (a) the peak values of V 1 and V 2 double. (b) the peak value of V 1 is reduced by a factor of two and V 2 doubles. (c) the peak value of V 1 does not change but V 2 doubles. (d) the peak value of V 1 does not change but V 2 is reduced by a factor of two. (e) the peak value of both V 1 and V 2 does not change.

5 14. (2 points) Unpolarized light of intensity I 0 passes through two Polaroid filters. The second filter has its axis inclined at an angle of 60 relative to the first, as shown below. The intensity of light exiting the second filter is: I 0 (a) 0.25I 0 (b) 0.375I 0 60 o object (a) f < 3.0 cm (b) f = 3.0 cm (c) f > 3.0 cm (d) There is not enough information to say anything about f. 17. (2 points) A red laser (of wavelength 700 nm) shines through a double slit and forms an interference pattern on a screen, as shown below. If we replace the red laser with a blue laser, what happens to the spacing of the bright fringes on the screen? 3.0cm (c) 0.50I 0 (d) 0.125I 0 (e) None of these 15. (2 points) A planar electromagnetic wave is propagating through space. Its electric field vector is given by E = E 0 cos(kx ωt) ĵ. Its magnetic field vector is (a) B = B 0 cos(kx ωt) î (b) B = B 0 cos(ky ωt) î (c) B = B 0 cos(kx ωt) ˆk (d) B = B 0 sin(kz ωt) ˆk (e) B = B 0 sin(kx ωt) ĵ 16. (2 points) A camera is focused on a nearby object. The object is imaged on the film, which is 3.0 cm behind the camera lens. What can you say about the focal length f of the lens? (a) They get closer together. (b) They get further apart. (c) The spacing doesn t change. 18. (2 points) Which type of electromagnetic wave travels through space the slowest? (a) radio waves (b) infrared light (c) visible light (d) X rays (e) All EM waves travel at the same speed.

6 Longer Questions: LITTLE OR NO CREDIT MAY BE EARNED FOR ANSWERS THAT DO NOT SHOW HOW YOU GOT THEM. Be sure to show your work. I will try to give you partial credit for correct steps if I can follow your logic. 19. (4 points) The wire shown carries a current I and consists of two very long straight segments connected by a perpendicular segment of length d. What is the magnitude of the B field a distance d directly below the bottom corner in the wire? d I I d B=? 20. (4 points) As shown in the figure below, a layer of water covers a slab of material X in a beaker. A ray of light traveling upward follows the path indicated. Using the information on the figure, find the index of refraction of material X, (the index of refraction for water is 1.33).

7 21. (6 points) A conducting disk of radius R is spinning counterclockwise with an angular velocity ω. A uniform magnetic field B is oriented perpendicular to the plane of the disk, as shown below. (a) (2 points) What direction (i.e., radially outward, radially inward) is the electric field pointing within the disk? (b) (4 points) What is the emf between the center and the outer edge of the disk?

8 Do TWO of the following THREE questions. CROSS OUT the question you do not want me to grade (otherwise I ll only grade the first two). 22. (8 points) In the circuit shown in the figure The circuit consists of three parallel branches. The left branch consists of a Ω resistor and a 75.0 V battery. The middle branch consists of a 5.0 mh and a 15.0 mh inductors. The right branch consists of a 35.0 µf and a 25.0 µf capacitors. Switch S connects the middle branch with the left branch (position 1) or the right branch (position 2). The capacitors are initially uncharged, and the switch has been in position 1 for a very long time. The switch is now suddenly flipped to position 2. (a) (2 points) What is the current flowing through the inductors right after the switch is flipped? (b) (2 points) What is the effective capacitance of the two capacitors in series? (c) (4 points) Find the maximum charge that the 25.0 µf capacitor will receive.

9 23. (8 points) A line charge is shaped as a quarter of a circle of radius R. The charge is distributed uniformly and has a net charge Q. (a) (5 points) Determine the magnitude of the electric field at the center. Express in terms of R, Q, and physical constants. (b) (3 points) If a proton (of charge e and mass m p ) were released from rest at the origin, how fast would be going very far from the semicircle (neglect gravity and other forces other than the electric force)?

10 24. (8 points) Coaxial Cable. A solid conductor with radius a is supported by insulating disks on the axis of a conducting tube with inner radius b and outer radius c in the following figure. The central conductor and tube carry equal currents I in opposite directions. The currents are distributed uniformly over the cross sections of each conductor. (a) (6 points) Derive an expression for the magnitude of the magnetic field, B(r), within the outer conducting tube (b < r < c). (b) (2 points) If the current I is increasing in time, in what direction is the induced electric field in the region between the conductors (for a < r < b)? (e.g., wrapping around inner conductor, parallel to inner conductor, radial, or is it zero)?

11 Equations k = 1 = Nm 2 /C 2 4πɛ 0 ɛ 0 = C 2 /Nm 2 µ 0 = 4π 10 7 Tm/A Mass of electron : m e = kg Mass of proton : m p = kg Charge of proton : e = C 1 ev = J F = qe k dq E = ˆr E da = Q enc ɛ 0 E l = dv dl r 2 V = V (r b ) V (r a ) = U = q V dq V = k r Q = C V V = IR Q = C V τ CR = RC U = 1 2 CV 2 R = ρ l A P = V I a c = v 2 /r rb r a E dl F = qv B F = Idl B µ0 I dl ˆr B = 4π r ( 2 ) dφ E B dl = µ 0 I enc + ɛ 0 dt B = µ 0I 2πr B = µ 0I 2r Φ B = B da E = dφ B = dt E dl L = Φ B /I U = 1 2 LI2 k = 2π/λ w = 2πf E 0 = cb 0 c = 1/ ɛ 0 µ 0 u B =.5B 2 /µ 0 u E =.5ɛ 0 E 2 u EB = ɛ 0 E 2 < u EB > = 1 2 ɛ 0E 2 E(x, t) = E 0 sin(kx wt) I = 0.5ɛ 0 E 2 0c n 1 sin(θ 1 ) = n 2 sin(θ 2 ) 1 s + 1 s = 1 f d sin θ = nλ