# 2006 #3 10. a. On the diagram of the loop below, indicate the directions of the magnetic forces, if any, that act on each side of the loop.

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5 # E3. A loop of wire of width w and height h contains a switch and a battery and is connected to a spring of force constant k, as shown above. The loop. carries a current I in a clockwise direction, and its bottom is in a constant, uniform magnetic field directed into the plane of the page. a. On the diagram of the loop below, indicate the directions of the magnetic forces, if any, that act on each side of the loop. b. The switch S is opened, and the loop eventually comes to rest at a new equilibrium position that is a distance x from its former position. Derive an expression for the magnitude B 0 of the uniform magnetic field in terms of the given quantities and fundamental constants. The spring and loop are replaced with a loop of the same dimensions and resistance but without the battery and switch. The new loop is pulled upward, out of the magnetic field, at constant speed v o. Express algebraic answers to the following questions in terms of B o, v o,, and the dimensions of the loop. c.i. On the diagram of the new loop below, indicate the direction of the induced current in the loop as the loop moves upward. ii. Derive an expression for the magnitude of this current. d. Derive an expression for the power dissipated in the loop as the loop is pulled at constant speed out of the field. e. Suppose the magnitude of the magnetic field is increased. Does the external force required to pull the loop at speed v o increase, decrease, or remain the same?. Increases Decreases emains the same

6 #1 12 Consider the electric field diagram above. a. Points A, B, and C are all located at y = 0.06 m. i. At which of these three points is the magnitude of the electric field the greatest? Justify your answer. ii. At which of these three points is the electric potential the greatest? Justify your answer. b. An electron is released from rest at point B. i. Qualitatively describe the electron's motion in terms of direction, speed, and acceleration. ii. Calculate the electron's speed after it has moved through a potential difference of 10 V. c. Points B and C are separated by a potential difference of 20 V. Estimate the magnitude of the electric field midway between them and state any assumptions that you make. d. On the diagram, draw an equipotential line that passes through point D and intersects at least three electric field lines.

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8 #3 16 A student performs an experiment to obtain the value of _ 0, the magnetic permeability of vacuum. She measures the magnetic field along the axis of the long, 100-turn solenoid PQ shown above. She connects ends P and Q of the solenoid to a variable power supply and an ammeter as shown. End P of the solenoid is taped at the 0 cm mark of a meterstick. The solenoid can be stretched so that the position of end Q can be varied. The student then positions a Hall probe* in the center of the solenoid to measure the magnetic field along its axis. She measures the field for a fixed current of 3.0 A and various positions of the end Q. The data she obtains are shown below. a. Complete the last column of the table above by calculating the number of turns per meter. * A Hall Probe is a device used to measure the magnetic field at a point. b. On the axes below, plot the measured magnetic field B versus n. Draw a best-fit straight line for the data points. c. From the graph, obtain the value of µ o, the magnetic permeability of vacuum. d. Using the theoretical value of µ o = 4! x 10-7 TM/A, determine the percent error in the experimental value of µ o computed in part (c).

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12 E&M 2 In the circuit shown above, resistors 1 and 2 of resistance 1 and 2, respectively, and an inductor of inductance L are connected to a battery of emf E and a switch S. The switch is closed at time t = 0. Express all algebraic answers in terms of the given quantities and fundamental constants. a. Determine the current through resistor 1 immediately after the switch is closed. E&M. 2. In the circuit shown above, A and B are terminals to which different circuit components can be connected. b. Determine the magnitude of the initial rate of change of current, di/dt, in the inductor. (a) Calculate the potential difference across 2 immediately after the switch S is closed in each of the following cases. i. A 50! resistor connects A and B. c. Determine the current through the battery a long time after the switch has been closed. ii. A 40 mh inductor connects A and B. d. On the axes below, sketch a graph of the current through the battery as a function of time. iii. An initially uncharged 0.80 µf capacitor connects A and B. (b) The switch gets closed at time t = 0. On the axes below, sketch the graphs of the current in the 100 W resistor 3 versus time t for the three cases. Label the graphs for the resistor, L for the inductor, and C for the capacitor. Some time after steady-state has been reached, the switch is opened. e. Determine the voltage across resistor 2 just after the switch has been opened.

13 Electricity and Magnetism Free esponse Answers 8.) 1980 E&M 3 a.) clockwise b.) I = " #r 2 C c.) E = rc 2 d.) E = a 2 C 2r 1.) 1992 E&M 3 a.) Out of the page. b.) i.) " m = µ 0#c 2\$ d.) The force is directed toward the wire. ( 1- %t & b ) ( ' a + ii.) E = µ 0"c# * 2\$ ln % b ( ' * & a ) c.) i.) counterclockwise ii.) counterclockwise 9.) 1985 E&M 3 a.) E = 47 V b.) E = 15 V m c.) v = 8.8 "10 6 m s d.) a = 2.6 "10 12 m s 2 2.) 1994 E&M 2 a.) i.) shuttle end ii.) V = 5016 V b.) I = A c.) i.) F = N ii.) to the left 10.) 2006 E&M 3 a.) b.) B 0 = kx Iw v 0 c.) i.) ii.) I ind = B 0w v 0 d.) "U = Pt = I 2 t = 1.52 #10 9 J e.) If the current was forced to flow the other way, the direction of the magnetic force on the current would be reversed. This force would do work on the shuttle, and the resulting gain in of energy would cause an increase in radius of the orbit. d.) P = B 2 0 w 2 2 v 0 I ind e.) Increases due to more induced current resulting in more force 3.) 1995 E&M 3 a.) E = BHv 0 b.) I = BHv 0 c.) e.) d.) # v = v 0 exp " B 2 H 2 t & % ( \$ m ' The current must be clockwise to create a magnetic field that opposes the applied field (Lenz s law). Upon entering the field (t = 0) the glider will slow down exponentially until the entire loop is in the field (t = t 1 ). At this time the glider will maintain a constant speed until the front edge of the loop begins to come out of the field (t = t 2 ). The glider will again start to slow down exponentially until the loop is completely out of the field (t = t 3 ) where the glider will once again move at a constant speed. 4.) 1984 E&M 3 a.) clockwise current b.) I = Blv 0 " " f.) show that E = # Pdt = I 2 dt = 1 2 # mv ) 1991 E&M 3 c.) left d.) F = B 2 l 2 v 0 e.) P = B 2 l 2 2 v # 0 exp " 2B 2 l 2 t & % ( \$ m ' a.) E = Blv 0 b.) I = Blv 0, counterclockwise c.) v = v exp # " B 2 l 2 t & 0 % ( d.) E = 1 \$ m ' 2 mv ) 1986 E&M 1 a.) All vectors drawn with correct sense (higher to lower potential) and all vectors drawn perpendicular to equipotential lines. b.) Magnitude of electric field is greatest at point T because equipotential lines are closest together near T. c.) E = "V "x = 500 V m f.) No, work does not depend upon path. 12.) 2005 E&M 1 d.) V M " V S = 35 V e.) W = q"v = 5#10 \$11 J a.) i.) point C where lines are closest ii.) point A The field points in the b.) i.) The electron moves to the together direction of decreasing potential. left with increasing speed. Magnitude of acceleration is decreasing. b.) ii.) v = 1.9 "10 6 m/s c.) E = 2000 V/m d.) curved line concave up must be perpendicular to the electric field lines 13.) 1989 E&M 1 a.) E = 0 b.) V = 0 c.) E = Q % 1 4"# 0 r \$ r ( ' * d.) V = Q % 1 & 2 3 ) 4"# 0 r + r 2 2 \$ 3 ( ' * & 3 2 ) 6.) 1987 E&M 1 a.) Q Qr Q E = b.) E = c.) V = 4"# 0 r 2 4"# 0 3 4"# 0 d.) V = 3Q 8"# E&M 2 a.) i.) I = Bhv ii.) F A = B 2 h 2 v b.) i.) Counterclockwise ii.) ight 7.) 1987 E&M 2 a.) " = 0.18e #4t b.) counterclockwise c.) i = 0.12e "4t d.) E = J Current I w 2w 3w 4w Force F x 5w w 2w 3w 4w 5w x

14 15.) 1983 E&M 1 22.) 1982 E&M 3 Q a.) E = b.) V 4"# 0 r 2 0 = Q b \$ a 4"# 0 ab c.) C = 4"# 0ab b \$ a a.) i A = E b.) i B = E 2 16.) 2005 E&M 3 a.) b.) graph B vs n and draw a line c.) find slope of the line of best fit and set equal of best fit using a straight edge to µ 0 I. Solve for µ 0 (I = 3.0 A). n d.) percent error = µ " µ 0 0ex ( 100) µ 0 c.) d.) Current starts at E and exponentially decays towards E as time approaches infinity. 2 E " 2i " L di dt = 0 e.) i( t) = E # # 2t && % 1+ exp %" (( 2 \$ \$ L '' 23.) 1998 E&M 2 a.) Draw meter connected parallel to 1. b.) V = 6.67 V c.) i.) V = 0 ii.) Q = 300 µc E d.) V = 0 e.) i.) I = = 0.67 A ii.) U L = J f.) E " I( ) " L di dt = 0 17.) 1998 E&M 3 a.) e.) I = mgsin" lb b.) v = mgsin" ( c.) P = m 2 g 2 sin 2 ") d.) v( t) = mgsin" \$ \$ 1# exp # B 2 l 2 t '' & B 2 l 2 B 2 l 2 B 2 l 2 & ) % % m ) (( Yes, the final speed of the bar decreases because the two resistors are in parallel across the emf in the bar and the new effective resistance is /2. This results in a speed reduction using the expression from (b). 24.) 2005 E&M 1 E di a.) I init = b.) dt = 2 E c.) I ( batt = E )L 1 d.) graph starts at initial I found in (a) and asymptotically approaches current finial I found in (c) concave down 18.) 1986 E&M 2 a.) I = 0.5 A b.) I = 0.5 A c.) Q = 100 µc d.) I = A e.) I L = A 19.) 1986 E&M 3 a.) B = µ 0ct 2"r # e.) F net = µ 2 0 c 2 b 2 t 4" 2 ln # a + b &&# % % (( 1 \$ \$ a '' a ) 1 & % ( \$ a + b ' 20.) 1991 E&M 2 b.) Counterclockwise c.) i = µ cb 0 2" ln # a + b & % ( d.) Away from the wire. \$ a ' a.) I i = 0 b.) I = 0.2 A c.) Current starts at 0 and approaches 0.2 A modified exponential growth d.) U L = 0.02 J e.) I = 0.2 A f.) V L = 30 V g.) The stored energy is dissipated in the resistor as thermal energy. e.) V 2 = E ) 2008 E&M 2 a.) i.) V 2 = 500 V ii.) V 2 = 900 V iii.) V 2 = 410 V b.) esistor graph: The current is constant with the value less than the initial value of the capacitor graph or the steady state value of the inductor graph. Inductor graph: The inductor initially opposes the flow of current, so the initial current in that branch is zero. Eventually, the inductor acts like a wire and does not impede the flow of charge, as the rate of change of current decreases to zero. The graph starts at I = 0 at time t = 0 and is concave down and asymptotic to the initial current in the capacitor case. Capacitor graph: Initially, the capacitor is uncharged and current is a maximum in the branch containing 3. As the capacitor charges the current in the branch decreases exponentially to zero concave up. 21.) 1987 E&M 3 a.) V 90" = 18 V b.) di dit = 36 A s c.) i = 2 A d.) U L = 1.0 J e.) V 90" = 180 V f.) Voltage starts at 180 V and exponentially decays towards zero as time approaches infinity.

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