{VERSION 2 3 "APPLE_PPC_MAC" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Autho r" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 "Computer Intensive Physic s" }}{PARA 18 "" 0 "" {TEXT -1 23 "Physics 212E: Fall 1997" }}}{EXCHG {PARA 19 "" 0 "" {TEXT -1 16 "Steven R. Dunbar" }}{PARA 19 "" 0 "" {TEXT -1 40 "Department of Mathematics and Statistics" }}{PARA 19 "" 0 "" {TEXT -1 30 "University of Nebraska-Lincoln" }}{PARA 19 "" 0 "" {TEXT -1 23 "Lincoln, NE, USA 68588" }}{PARA 19 "" 0 "" {TEXT -1 20 " sdunbar@math.unl.edu" }}{PARA 0 "" 0 "" {TEXT -1 32 "http://www.math.u nl.edu/~sdunbar" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "\n11-1: Fara day's Law, Induced EMFs\n\nKEYWORDS: Faraday's Law, Induced EMF, Magne tic Flux, Current Changing in Time." }}{PARA 0 "" 0 "" {TEXT -1 315 " \+ \nOBJECTIVE: Calculate a spatially varying magnetic flux through curre nt loops arranged in simple geometrical shapes, The current creating \+ the spatially varying magnetic field is varying in time, thereby creat ing an induced emf by Faraday's law. The current resulting from the e mf is calculated using Ohm's law. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 141 "Write Faraday's law e= - d\\Phi/dt, def ine all symbols appearing in it, and apply the sign convention relatin g the directions of Phi and e. " }}{PARA 0 "" 0 "" {TEXT -1 90 "\nPR EREQUISITE: Understand basic vector calculus, including flux integrals . Know Ohm's law" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 44 "LENGTH: One class period, about 50 minutes." }}{PARA 0 "" 0 "" {TEXT -1 59 "AUDIENCE: Instructor, student use in computer \+ lab setting." }}{PARA 0 "" 0 "" {TEXT -1 21 "SOFTWARE: Maple V R 4" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "FUTURE: \+ Need some good drawings instead of the ascii drawings. Need a good d rawing of the ciruclar loop problem." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "HISTORY: Created November 6, 1997" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "SOURCE: \+ Adapted, revised, and extended from Sherwood and Chabay, Chapter 13, H W13-4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Suggested InfoMall Reading: " }}{PARA 0 "" 0 "" {TEXT -1 73 "Suggested Keywords: Faraday's law, induced emf, induction, mag netic flux" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 73 "Problem I: Time-Varying Current in Long, Straight Wire, \+ Rectangular Loop" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 371 "A very long \+ wirre carries a current I1 upward, as shown, and this current is decre asing with time. Nearby is a rectangular loop of wire lying in the pl ane of the long wire, and containing a resistor R; the resistance of t he rest of the loop is negligible compared to R. The loop has a width w and a height h, and is located a distance d to the right of the l ong wire." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 " | -------d--------| --------------------- w ------------------- ---|" }}{PARA 0 "" 0 "" {TEXT -1 4 " |" }}{PARA 0 "" 0 "" {TEXT -1 82 " | +-------------------------------------- --------+ ---" }}{PARA 0 "" 0 "" {TEXT -1 100 " | \+ | \+ | |" }}{PARA 0 "" 0 "" {TEXT -1 99 " | \+ | < \+ |" }}{PARA 0 "" 0 "" {TEXT -1 96 " | | \+ R > h" }} {PARA 0 "" 0 "" {TEXT -1 99 " | | \+ < |" }}{PARA 0 "" 0 "" {TEXT -1 100 " | | \+ | |" }}{PARA 0 "" 0 " " {TEXT -1 82 " | +--------------------------- -------------------+ ---" }}{PARA 0 "" 0 "" {TEXT -1 8 " | I1" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 273 "First, calculate the magnetic field \+ at a point x units to the right of the long, straight, vertical wire, \+ due to the current I1 in the wire. Explain why the magnetic field nee d only be considered at a distance x to the right, and why the magneti c field does not vary in y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7 "B1 := ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Explanation: " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "Next, calculate the element of flux (vector flow ac ross an area) in a vertical strip of width dx at the distance x." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dPhi := ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Explicitly select out the integrand from the express ion for the flux element." }}{PARA 0 "" 0 "" {TEXT -1 149 "(Remember: \+ Maple can integrate the the expression for the integrand only. Maple can't integrate the heuristic element we create with the dx in it!)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "flux_integrand := ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 91 "Next calculate the total, or net, flux t hrough the entire rectangular loop at a given time." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Phi := ;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Now, explicitly r ecognize that current I1 is a function of time, and substitute that in , and reassign the result to Phi" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Phi := subs(I1 = I1(t), \+ );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The time rate of change of the flux gives the magnitude of the \+ current" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "emf := ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 52 "Now apply Ohm's law to find the current i n the loop." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "I2 := ; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "normal(I2);" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 71 "Problem II: Time-Varying Current in Long , Straight Wire, Circular Loop" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 361 "A very long wirre carries a current I1 upward, as shown, and this current is decreasing with time. Nearby is a circular loop of wire l ying in the plane of the long wire, and containing a resistor R; the r esistance of the rest of the loop is negligible compared to R. The lo op has a radius d and a center located a distance d to the right o f the long wire." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "First, calculate \+ the magnetic field at a point x units to the right of the long, straig ht, vertical wire, due to the current I1 in the wire. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "B1 :=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "Next, calculat e the element of flux (vector flow across an area) in a vertical strip of width dx at the distance x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "assume(d>0);\ndPhi := ;" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "E xplicitly select out the integrand from the expression for the flux el ement." }}{PARA 0 "" 0 "" {TEXT -1 149 "(Remember: Maple can integrat e the the expression for the integrand only. Maple can't integrate th e heuristic element we create with the dx in it!)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "flux_integr and := ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Next calculate the total, or net, flux through the entire circular loop at a given time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "normal(flux_integral); # cl ean up the algebra, etc." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " Phi := ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Now, explicitly recognize that current I1 is a function \+ of time, and substitute that in, and reassign the result to Phi" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Phi := subs(I1 = I1(t), );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The time rate of change of the \+ flux gives the magnitude of the current" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "emf := ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Now apply Ohm's law to find the current in the loop." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "I2 := ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "normal(I2); # more clean up if possible" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{MARK "2 0 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 }