{VERSION 2 3 "APPLE_68K_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 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 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 "T itle" 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 "Author" 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 -1 263 "\n1-1: Kirchh off's Laws. and Direct Current Circuits, \n\n\nKEYWORDS: Kirchhoff's L aws, Direct-Current Circuits, Reisistance in Series and Parallel Circu its, Equivalent Reisitance\n\nOBJECTIVE: Derive the equivalent resista nce of circuits with multiple loops and nodes." }}{PARA 0 "" 0 "" {TEXT -1 90 "\nPREREQUISITE: Know Ohm's law, basics of solving multipl e equations in multiple unknowns.\n" }}{PARA 0 "" 0 "" {TEXT -1 47 "LE NGTH: One class period, about 40-50 minutes." }}{PARA 0 "" 0 "" {TEXT -1 59 "AUDIENCE: Instructor, student use in computer lab settin g." }}{PARA 0 "" 0 "" {TEXT -1 21 "SOFTWARE: Maple V R 4" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "FUTURE: Add the \+ figures of the circuits, and also add some further explanations and ti ps and pointers on setting and solving the algebraic signs of the curr ents." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 " HISTORY: Created October 1, 1997" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Suggested InfoMall Reading: Dudl ey Williams and John Spangler, Physics for Science and Engineering," } }{PARA 0 "" 0 "" {TEXT -1 128 "Sections 26.2, 26.4; and E.M. Rogers, \+ Physics for the Enquiring Mind. Suggested Keywords: Resistance and S eries and Parallel." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 21 "Defiintions and Laws:" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 257 11 "Definition:" }{TEXT -1 4 " A " }{TEXT 256 12 "branch \+ point" }{TEXT -1 77 " in a circuit network is a place where three (or \+ more) conductors are joined." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 258 12 "Definition: " }{TEXT -1 3 " A " } {TEXT 259 4 "loop" }{TEXT -1 52 " (also called a mesh) is any closed c onducting path." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 18 "Kirchhoff's Law 1:" }{TEXT -1 80 " (Point Rule) T he algebraic sum of the currents toward a branch point is zero." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 17 " Kirchhoff's Law 2" }{TEXT -1 122 ": (Loop Rule) The algebraic sum of the voltages (emf's) in any loop equals the sum of the I*R products i n the same loop." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 262 7 "Step 1:" }{TEXT -1 109 " Assign a symbol and a di rection to all unknown emfs and currents, and a symbol to all unknown \+ resistances. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 7 "Step 2:" }{TEXT -1 171 " Represent all in a diagram wih all directions carefully shown. Imagine the network to be \"sepa rated\" into a number of separate loops like the pieces of a jigsaw pu zzle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 7 "Step 3:" }{TEXT -1 217 " If there are n branch points in the network, apply the Law 1 (Point Rule) at n-1 of these points. An y points may be chosen. Application of the point rule at the n-th poi nt does not lead to an independent equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 7 "Step 4:" }{TEXT -1 76 " Apply the loop rule to each of the loops from the decomposition of S tep 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 72 "Example 1: Hands-On Activity #1: A Circuit with TwoLoops , Two Batteries" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 173 "Consider the circuit in the \+ Hands-On Activity #1. (See the diagram for a reminder.) The emf's an d resistances are known. Find the currents in each section of the cir cuit." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " eqn_pt_d := -I1 + I2 + I3 = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn_loop_1 := E1 = I1*R1 + I3*R3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " eqn_loop_2 := E2 = I2*R2 + I3*R3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eqns := \{eqn_pt_d, eqn_loop_1, eqn_loop_2\};" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "unknowns := \{I1, I2, I3\}; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "currents := solve( eqns , unknowns);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "subs( \{E1 \+ = 1.5*V, E2 = 3.0*V, R1 = 6*Ohm, R2 = 7*Ohm, R3 = 50*Ohm\}, currents); " }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 67 "Example 2: Hands-On Activity #2: A Wheatstone-Bridge-Like Circuit" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn_pt_a := I6 - I1 - I3 = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn_pt_b := I1 - I2 - I5 = 0; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn_pt_d := I5 + I3 - I4 = 0;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "eqn_loop_1 := I1*R1 + I5*R5 - I3*R3 = 0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqn_loop_2 := I2*R2 - I4*R4 \+ - I5*R5 = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn_loop_3 \+ := I3*R3 + I4*R4 = E1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "e quations := \{eqn_pt_a, eqn_pt_b, eqn_pt_d, eqn_loop_1, eqn_loop_2, eq n_loop_3\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "unknowns := \+ \{I1,I2, I3, I4, I5, I6\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "currents := solve(equations, unknowns);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "subs( \{R1 = 10, R2 = 10, R3 = 10, R4 = 10, R5 = 5 0\}, currents);" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 110 "Example 3: P roblem Place (Schaum's Worked Examples) 27.138, Equivalnet Resistance \+ Through a Cube of Resistors" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "eqn1 := i = i1 + i4 + i5;\neqn2 := i1 = i2 + i6;\neqn3 := i2 + i3 = i7;\neqn4 := \+ i4 = i3 + i8;\neqn5 := i5 = i9 + i12;\neqn6 := i9 + i6 = i10;\neqn7 := i12 + i8 = i11; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "loop0 := -i1*r1 - i2*r2 + i3*r3 + i4*r4 = 0;\nloop1 := i1*r1 +i6*r6 -i9*r9 \+ -i5*r5 = 0;\nloop2 := -i4*r4 + i5*r5 + i12*r12 -i8*r8 = 0;\nloop3 := i 11*r11 -i7*r7 - i3*r3 + i8*r8 = 0;\nloop4 := -i10*r10 - i6*r6 +i2*r2 + i7*r7 = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "eqns := \{eq n1, eqn2, eqn3, eqn4, eqn5, eqn6, eqn7, loop0, loop1, loop2, loop3, lo op4\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "unknowns := \{i1, i2,i3,i4, i5, i6, i7, i8, i9,i9, i10, i11, i12\};" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "solve(eqns, unknowns);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "currents := \":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "r1 := r;\nr2 := r;\nr3 := r;\nr4 := r;\nr5 :=r;\nr6 \+ := r;\nr7 := r;\nr8 := r;\nr9 := r;\nr10 := r;\nr11 := r;\nr12 := r;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "currents;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "emf_eqn := i*Req = i1*r1 + i2*r2 + \+ i7*r7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "emf_eqn := subs( \+ currents, emf_eqn);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solv e( emf_eqn, Req);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "2 9 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }