Name: Student number: Academic Honesty: Cheating in an examination includes the following: 1. the unauthorized sharing of material such as textbooks during an open book examination; 2. concealing information pertaining to the examination in the examination room, or in washrooms or other places in the vicinity of the examination room; 3. using course notes or any other aids not approved by an nstructor during an examination; or, 4. the unauthorized possession or use of an examination question sheet, an examination answer book, or a completed examination or assignment. Question # Mark Maximum Mark Multiple Choice 20 11 10 12 10 13 10 14 10 Total= /60 Enter your answers to the multiple choice questions here by blackening in the circle corresponding to the best answer. There is only one answer per question. 1. A B C D E 2. A B C D E 3. A B C D E 4. A B C D E 5. A B C D E 6. A B C D E 7. A B C D E 8. A B C D E 9. A B C D E 10. A B C D E 1
There are 10 multiple choice questions. Select the correct answer for each one and mark it on the bubble form. Each question has only one correct answer. (2 marks each) 1. A constant current,, is supplied for a brief time to charge a parallel plate capacitor. The capacitor has circular plates of radius R with a gap d (d << R). A B C D At which of the positions A, B, C or D is the magnetic field the smallest? (a) A (b) B (c) C Correct (d) D (e) The magnetic field is zero at all of A, B and C. When a battery is in use current flows through the small internal resistor r causing the voltage across the external terminals to be less than the ideal voltage. The battery connected to two 1R resistors draws more current and thus reduces the output voltage of the battery more than the one connected to a singl 1R resistor. 3. Two long straight current-carrying parallel wires cross the x axis and carry currents and 3 in the same direction, as shown. At what value of x is the net magnetic field zero? Point C is at the centre of the capacitor plates, so the area enclosed by the circular pathlength through C around the centre is zero: Φ E = 0. Therefore, of the circulation calculation B d l = 0. Since B must be equal in magnitude given the symmetry of the capacitor, B = 0. Other points have nonzero pathlengths that enclose changing electrical flux or enclose current. 2. The real batteries and the resistors in both cases illustrated above are identical. n which case is the voltage across the terminals of the battery closest to the ideal battery voltage V 0? (a) 0 (b) 1m (c) 3 m (d) 5m correct (e) 7m The magnetic field scales as /R so (a) The voltage across the terminals of the battery is closest to ideal in Case A. Correct (b) The voltage across the terminals of the battery is the same in both cases. (c) The voltage across the terminals of the battery is closest to ideal in Case B /3 = R 2 /R 1 = (D x)/x where x is the distance from the 1 mark on the ruler. D = 4 so x = 3. The 5 position is 3 times farther from the left-hand wire that carries 1/3 the current of the right-hand one. 2
4. Consider a long wire running in the vertical direction with a rectangular loop of wire beside it as shown. Which of the following situations would result in a clockwise induced current in the loop? (a) A current in the long wire, directed upward, is increasing in magnitude. (b) With a constant current in the long wire directed upward, the loop is moved toward the top of the page, parallel to the long wire. (c) The loop is held stationary and the long wire, while carrying a constant upward current, is moved away from the loop. Correct The upward current in the long wire results in a magnetic field through the down directed into the page. By moving the wire farther from the loop, there is a reduction in the downward B field through the loop, so a clockwise induced current would serve to inhibit the decrease in B through the loop as required b Lenz s law. 5. A long, straight wire of radius a carries a steady current. The graph of magnetic field strength B(r) as a function of perpendicular distance r from the centre of the wire is: nside the wire, the circle passing through position r encloses more and more current inside, implying larger magnetic field at that point. When r increases outside the wire, the total current enclosed remains the same which by Ampere s law implies that B decreases as 1/r. 3
The next two questions relate to the situation described below. Which of the three emerging beams contains the isotope of the largest mass assuming the ions are equally charged? A tiny wire loop in the xz plane carries a current and is inside a larger loop in the xy plane as shown. Both carry a current. z positive ion beam magnet A BC x 6. What is the direction of the magnetic moment of the small loop? (a) ˆx (b) ˆx (c) ŷ Correct (d) ẑ (e) ẑ Curl the fingers of your right hand in the direction of the current and your thumb points in the ŷ direction. 7. What is the direction of the torque on the small loop? (a) ˆx (b) ˆx (c) ŷ (d) ẑ (e) ẑ The lowest energy occurs when the magnetic moments of the two loops align so the torque will flip the small loop around the x axis, clockwise when looking down the x axis when viewed looking towards the origin from its positive end. Curling your fingers around the x axis in the direction of the flip, and the thumb points in the ˆx direction. 8. Singly charged ions are projected with the same speed through a region of magnetic field in which they are bent in a 90 arc. sotopes of different masses are separated during their trajectory. y on source Beam A must have the largest mass because it s radius is larger. t takes more force to bend a beam of particles whose mass is larger, but charge and velocity are the same. 9. n the Faraday s Law experiment you did in class, which of the following describes your qualitative results when the oscilloscope channel 1 displays the voltage across the resistor attached to the field coil and channel 2 displays the emf from the pickup coil. (a) When channel 1 displays a square wave, channel 2 displays a square wave (b) When channel 1 displays a square wave, channel 2 displays a triangle wave Correct (c) When channel 2 displays a square wave, channel 1 displays a triangle wave (d) When channel 2 displays a sine wave, channel 1 displays a square wave (e) When channel 1 displays a sine wave, channel 2 displays a triangle wave Channel 2 must have the shape of the derivative of the signal in channel 1. 10. A capacitor, in series with a resistor, is being charged. At the end of 10 ms the charge on its positive plate is half the final value. The time constant for the process is about (a) 0.43 ms (b) 2.3 ms (c) 6.9 ms (d) 10 ms (e) 14 ms 4
Let t h represent the half life and τ the time constant. V(t) = V 0 (1 e t h/τ ) = 0.5V 0 e t h/τ = (1 0.5) t h /τ = ln(0.5) τ = 10/ ln(0.5) = 14.4 s There are four written problems. Show all your work to get full credit. and give answers to at least 3 significant figures. Consider all values exact 11. n the following circuit V 1 = 20 V, V 2 = 4.5 V, R 1 = 5 ohm, R 1 = 5 ohm, R 2 = 10 ohm, R 3 = 15 ohm, R 4 = 20 ohm. (a) Use Kirchhoff s rules to find equations which will allow determining the currents 1, 2 and 3. : V1 V2 1*R4 = 0 : 1*R4-3*R2 4*R1 = 0 : V1 5*R3 3*R2 = 0 V: V2-4*R1-5*R3 V: 1 3-2 = 0 V: 5-4 - 3 = 0 V 1 2 4 3 5 V Three of (,,, and V) plus V and V can give a complete solution. 4 and 5 have to be inserted by the student and may be different from what is done here. Allow for that when grading. (b) t turns out that 3 = 1.7 A. f so find the potential difference between points a and b. V ab = 1.7 10 = 17 V V 5
12. A wire of length 10 cm and mass 10 g, and resistance 0.10 ohm is constrained to fall while in contact with a u-shaped conducting frame as shown in the figure. There is a uniform magnetic field of 0.10 T directed horizontally, perpendicular to the falling wire. (a) f the rod is falling with speed v, what is the EMF induced in the loop? The rate of change of the area in the loop is da/dt = lv. Φ B dt = Blv = E Where l is the length of the wire or the width of the U-frame. (b) At speed v what is the Lorentz force on the wire due to the magnetic field? F = l B Where l is the length of the wire (or the width of the U-frame. Calculating magnitudes = E/R F = B 2 l 2 v/r (b) Calculate the steady-state velocity of the wire, v t. Terminal velocity v t occurs when F = m g Therefore F = B 2 l 2 v t /R = mg Which turns out to be 98 m/s. v t = Rmg B 2 l 2 6
13. n the series LRC circuit shown, L = 0.10 mh and R = 100 ohms and C = 10 µ F. (a) Draw the phasor diagram for the resistances, reactances and total impedance when ω = 2000 rad/s. X L R R X C Z total L C Phasor Diagram (b) What frequency ω (in rad/s) is necessary in order for the voltage amplitudes measured across the resistor and across the capacitor to be equal? ω = 1/RC = 1000 rad/sec (c) At this frequency, what is the phase shift, φ between V C and V R. Specify whether V C leads or lags V R. 90 deg or π/2. V R leads V C. (d) What is the resonant frequency of the circuit in Hz? ω 0 = 1/ LC, f = ω/2π = 5032 Hz. 5000 Hz is close enough. (e) How much energy is dissipated in the resistor per cycle at resonance? Energy in one cycle is PT = E 2 rms/r f = E 2 /2R f where T is the period and f the frequency. The value of E isn t specified so symbolic answer is good enough. F they use E then they should divide by 2. 7
14. A plane electromagnetic wave in a vacuum is given by E = 5( î ĵ) sin(kz ωt) V/m B = B 0 sin(kz ωt) where k = 2π λ and ω = 4π 1015 rad/s (a) What is E 0? E 0 = 5 2 V/m (b) n what direction does the wave propagate? How do you know? t propagates in the z direction because kz ωt has the same sign in both terms. (c) What is B 0? (magnitude and direction) B 0 = 5( î ĵ)/c = (5/3)( î ĵ) 10 8 T (this is good enough. Or B 0 = 5 2/c T at -45.) (d) What is the wavelength, λ, in metres? λ = 2πc/ω = 1.5 10 7 m (e) Describe the polarization state as accurately as you can. (Specify angles from an axis if linear, or handedness if circular or elliptical. Draw a well-labelled picture.) Linear polarized 135 degrees from the x axis. 8