{VERSION 3 0 "DEC ALPHA UNIX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 88 " \+ Radiation From an Infinitely Long Wire" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 " \+ Ross L. Spencer, Brigham Young Univ ersity" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "This Maple worksheet finds the electromagnetic field produced by an infinitely long wire carrying current " } {XPPEDIT 18 0 "I[0]*cos(omega*t);" "6#*&&%\"IG6#\"\"!\"\"\"-%$cosG6#*& %&omegaGF(%\"tGF(F(" }{TEXT -1 20 " in the z-direction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "In this situation there is no char ge density " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 26 " but only \+ current density " }{XPPEDIT 18 0 "J[z] = I[0]*cos(omega*t)*delta*x*del ta(y);" "6#/&%\"JG6#%\"zG*,&%\"IG6#\"\"!\"\"\"-%$cosG6#*&%&omegaGF-%\" tGF-F-%&deltaGF-%\"xGF--F46#%\"yGF-" }{TEXT -1 90 " . Putting this cu rrent density into the retarded-time form of the vector potential give s" }}{PARA 0 "" 0 "" {TEXT -1 59 " \+ " }{XPPEDIT 18 0 "A[z](x,y,z,t) = mu[0]*int(I[0 ]*cos(omega*(t-t[r]))/abs(r-rp),zp = -infinity .. infinity)/(4*Pi);" " 6#/-&%\"AG6#%\"zG6&%\"xG%\"yGF(%\"tG*(&%#muG6#\"\"!\"\"\"-%$intG6$*(&% \"IG6#F1F2-%$cosG6#*&%&omegaGF2,&F,F2&F,6#%\"rG!\"\"F2F2-%$absG6#,&FBF 2%#rpGFCFC/%#zpG;,$%)infinityGFCFMF2*&\"\"%F2%#PiGF2FC" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 59 " is the vector position of the observation point and whe re " }{XPPEDIT 18 0 "rp;" "6#%#rpG" }{TEXT -1 107 " stands for the vec tor r-prime, the position of the source point. In this cylindrical ge ometry the vector " }{XPPEDIT 18 0 "r;" "6#%\"rG" }{TEXT -1 46 " point s straight out from the wire a distance " }{XPPEDIT 18 0 "s;" "6#%\"sG " }{TEXT -1 23 " and the source vector " }{XPPEDIT 18 0 "rp;" "6#%#rpG " }{TEXT -1 199 " points along the wire in the z-direction. The retar ded time is just the current time t minus the delay time due to the sp eed of light between the source point at rp and the observation point \+ at r: " }{XPPEDIT 18 0 "abs(r-rp)/c;" "6#*&-%$absG6#,&%\"rG\"\"\"%#rpG !\"\"F)%\"cGF+" }{TEXT -1 58 " . Since the r and rp vectors are perp endicular we have " }{XPPEDIT 18 0 "abs(r-rp) = sqrt(zp^2+s^2);" "6#/- %$absG6#,&%\"rG\"\"\"%#rpG!\"\"-%%sqrtG6#,&*$%#zpG\"\"#F)*$%\"sG\"\"#F )" }{TEXT -1 66 " , and since everything is uniform in z the integral \+ above becomes" }}{PARA 0 "" 0 "" {TEXT -1 58 " \+ " }{XPPEDIT 18 0 "A[z](s,t) = mu[0]* int(I[0]*cos(omega*t-omega*abs(zp^2+s^2)/c)/sqrt(zp^2+s^2),zp = -infin ity .. infinity)/(4*Pi);" "6#/-&%\"AG6#%\"zG6$%\"sG%\"tG*(&%#muG6#\"\" !\"\"\"-%$intG6$*(&%\"IG6#F0F1-%$cosG6#,&*&%&omegaGF1F+F1F1*(F>F1-%$ab sG6#,&*$%#zpG\"\"#F1*$F*\"\"#F1F1%\"cG!\"\"FJF1-%%sqrtG6#,&*$FE\"\"#F1 *$F*\"\"#F1FJ/FE;,$%)infinityGFJFVF1*&\"\"%F1%#PiGF1FJ" }{TEXT -1 4 " \+ ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "We now simplify this integral in the following ways. (i) Since the inte grand is even in " }{XPPEDIT 18 0 "zp;" "6#%#zpG" }{TEXT -1 156 " we c an multiply by 2 and change the limits to go from 0 to infinity. (ii) We try to get rid of the ugliness in the integrand by making the u-su bstitution " }{XPPEDIT 18 0 "sqrt(zp^2+s^2) = u*s;" "6#/-%%sqrtG6#,&*$ %#zpG\"\"#\"\"\"*$%\"sG\"\"#F+*&%\"uGF+F-F+" }{TEXT -1 72 " . Making this change of limits and integration variable then leads to" }} {PARA 0 "" 0 "" {TEXT -1 58 " \+ " }{XPPEDIT 18 0 "A[z](s,t) = mu[0]*I[0]*int(cos(omeg a*t-omega*s*u/c)/sqrt(u^2-1),u = 0 .. infinity)/(2*Pi);" "6#/-&%\"AG6# %\"zG6$%\"sG%\"tG**&%#muG6#\"\"!\"\"\"&%\"IG6#F0F1-%$intG6$*&-%$cosG6# ,&*&%&omegaGF1F+F1F1**F>F1F*F1%\"uGF1%\"cG!\"\"FBF1-%%sqrtG6#,&*$F@\" \"#F1\"\"\"FBFB/F@;F0%)infinityGF1*&\"\"#F1%#PiGF1FB" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "Now \+ look, something interesting has already shown up. We expect to get ra dially outgoing waves at frequency " }{XPPEDIT 18 0 "omega;" "6#%&omeg aG" }{TEXT -1 46 ", so we might expect to see something like " } {XPPEDIT 18 0 "cos(k*s-omega*t);" "6#-%$cosG6#,&*&%\"kG\"\"\"%\"sGF)F) *&%&omegaGF)%\"tGF)!\"\"" }{TEXT -1 25 " showing up. But since " } {XPPEDIT 18 0 "omega = k*c;" "6#/%&omegaG*&%\"kG\"\"\"%\"cGF'" }{TEXT -1 29 " this factor would look like " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "cos(omega*s/c-omega*t);" "6#-%$cosG6#,&*(%&omegaG\"\"\"%\"sGF)%\"cG! \"\"F)*&F(F)%\"tGF)F," }{TEXT -1 187 " , which is very much like the n umerator of the integrand. Now you can see how traveling waves come \+ out of this retarded-time formalism: the retarded time contribution j ust turns into " }{XPPEDIT 18 0 "k*s;" "6#*&%\"kG\"\"\"%\"sGF%" } {TEXT -1 232 ". You are already familiar with this effect if you have ever made \"waves\" with a garden hose. As you wiggle the nozzle up \+ and down in time the retarded-time effects in the flow turn your time \+ oscillation into a sine wave in space." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 188 "Ok, back to the math. It is now tim e to try to do this integral. Maple seems not to know how to do it, b ut both Abramowitz and Stegun and Gradshteyn and Ryzhik give these use ful formulas" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 " \+ " }{XPPEDIT 18 0 "int(sin(u*alpha)/sqrt(u^2-1),u = 1 .. infini ty) = Pi*J[0](alpha)/2;" "6#/-%$intG6$*&-%$sinG6#*&%\"uG\"\"\"%&alphaG F-F--%%sqrtG6#,&*$F,\"\"#F-\"\"\"!\"\"F6/F,;\"\"\"%)infinityG*(%#PiGF- -&%\"JG6#\"\"!6#F.F-\"\"#F6" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " " }} {PARA 0 "" 0 "" {TEXT -1 71 " \+ " }{XPPEDIT 18 0 "int(cos(u*alpha)/sqrt( u^2-1),u = 1 .. infinity) = -Pi*Y[0](alpha)/2;" "6#/-%$intG6$*&-%$cosG 6#*&%\"uG\"\"\"%&alphaGF-F--%%sqrtG6#,&*$F,\"\"#F-\"\"\"!\"\"F6/F,;\" \"\"%)infinityG,$*(%#PiGF--&%\"YG6#\"\"!6#F.F-\"\"#F6F6" }{TEXT -1 3 " ," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "J[0];" "6#& %\"JG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Y[0];" "6#&%\"YG6#\" \"!" }{TEXT -1 139 " are the usual Bessel functions. Expanding the co sine in our integral and using these two identities then gives the fol lowing formula for " }{XPPEDIT 18 0 "A[z];" "6#&%\"AG6#%\"zG" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 61 " \+ " }{XPPEDIT 18 0 "A[z](s,t) = mu[0]*I[ 0]*(J[0](k*s)*sin(omega*t)-Y[0](k*s)*cos(omega*t))/4;" "6#/-&%\"AG6#% \"zG6$%\"sG%\"tG**&%#muG6#\"\"!\"\"\"&%\"IG6#F0F1,&*&-&%\"JG6#F06#*&% \"kGF1F*F1F1-%$sinG6#*&%&omegaGF1F+F1F1F1*&-&%\"YG6#F06#*&F=F1F*F1F1-% $cosG6#*&FBF1F+F1F1!\"\"F1\"\"%FN" }{TEXT -1 4 " ." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Now that we have this f ormula Maple can do us some good." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 1 0 4 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Az: =mu[0]*Io*(BesselJ(0,k*s)*sin(omega*t)-BesselY(0,k*s)*cos(omega*t))/4; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We get the electric field fro m " }{XPPEDIT 18 0 "E[z] = -diff(A[z],t);" "6#/&%\"EG6#%\"zG,$-%%diffG 6$&%\"AG6#F'%\"tG!\"\"" }{TEXT -1 31 " and the magnetic field from \+ " }{XPPEDIT 18 0 "B = curl(A);" "6#/%\"BG-%%curlG6#%\"AG" }{TEXT -1 6 " , or " }{XPPEDIT 18 0 "B[phi] = -diff(A[z],s);" "6#/&%\"BG6#%$phiG,$ -%%diffG6$&%\"AG6#%\"zG%\"sG!\"\"" }{TEXT -1 3 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Ez:=-diff(Az,t);Bphi:=-diff(Az,s);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "To see what's going on, let's make some p lots. " }{XPPEDIT 18 0 "E[z];" "6#&%\"EG6#%\"zG" }{TEXT -1 21 " will be in red and " }{XPPEDIT 18 0 "B[phi];" "6#&%\"BG6#%$phiG" }{TEXT -1 32 " will be in blue at a time when " }{XPPEDIT 18 0 "omega*t = Pi/ 4;" "6#/*&%&omegaG\"\"\"%\"tGF&*&%#PiGF&\"\"%!\"\"" }{TEXT -1 35 " so that we won't have either the " }{XPPEDIT 18 0 "cos(omega*t);" "6#-%$ cosG6#*&%&omegaG\"\"\"%\"tGF(" }{TEXT -1 5 " or " }{XPPEDIT 18 0 "sin (omega*t);" "6#-%$sinG6#*&%&omegaG\"\"\"%\"tGF(" }{TEXT -1 104 " term s missing from the formulas. And just to get a picture out we will se t all of the constants to 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "omega:=1;t:=Pi/4;mu[0]:=1;Io :=1;c:=1;k:=omega/c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plo t([Ez(s),Bphi(s)],s=0..50,-1..1,color=[red,blue],numpoints=200);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "As you can see from the plot, the re is some complicated stuff going on near the wire, but further out w e just have an outgoing wave with " }{XPPEDIT 18 0 "E[z];" "6#&%\"EG6# %\"zG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "B[phi];" "6#&%\"BG6#%$phiG " }{TEXT -1 104 " in phase, but with opposite signs. This is signific ant because the Poynting vector is proportional to " }{XPPEDIT 18 0 "c ross(E[z],B[phi]);" "6#-%&crossG6$&%\"EG6#%\"zG&%\"BG6#%$phiG" }{TEXT -1 34 ", which is in the -s direction if " }{XPPEDIT 18 0 "E[z];" "6#& %\"EG6#%\"zG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B[phi];" "6#&%\"BG6# %$phiG" }{TEXT -1 267 " have the same sign. With opposite signs we ha ve outgoing energy, as we should. You will also notice that the ampli tude of the waves are going down as we go out. Mathematically this is because at large x the Bessel functions turn into sines and cosines d ivided by " }{XPPEDIT 18 0 "sqrt(x);" "6#-%%sqrtG6#%\"xG" }{TEXT -1 44 " . So that's how the fields fall off: like " }{XPPEDIT 18 0 "1/sq rt(s);" "6#*&\"\"\"\"\"\"-%%sqrtG6#%\"sG!\"\"" }{TEXT -1 195 " . This is significant too, because this means that the Poynting vector falls off like 1/s. The energy flowing through a cylindrical surface of ra dius s centered on the wire is proportional to " }{XPPEDIT 18 0 "2*pi* s;" "6#*(\"\"#\"\"\"%#piGF%%\"sGF%" }{TEXT -1 102 ", so the total ener gy flowing out, which is proportional to the product of S and the area is constant." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 208 "To see the difference between the far-field region where we just have outgoing waves and the near-field region where the field is like the field of an oscillating current in a wire, let's use an a nimation of " }{XPPEDIT 18 0 "B[phi];" "6#&%\"BG6#%$phiG" }{TEXT -1 170 " (note: after the picture comes up, click on it and use the play er controls on the toolbar. I suggest that you set it for continuous \+ loop and slow it down quite a bit.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(plots):unassign('t');" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "animate( Bphi,s=.2..20,t=0..6*Pi,numpoints=100,frames=50,labels=[\"s\",\"Bphi\" ],color=navy);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "It is possible to do even more with the Poynting ve ctor. It is given by (we first have to unassign all of the constants \+ we gave special values)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "unassign('mu[0]','Io','omega','t','k');" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "S:=-Ez*Bphi/mu[0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(S);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 588 "These terms come in two types: (i) terms with cosines or sines squared and (ii) terms with products of sines and cosines. \+ This time dependence makes S wiggle around in time, sometimes positive , sometimes negative, so the energy doesn't just go out, but sometimes comes in as well. It is like ballroom dancers spinning across the fl oor. Overall their kinetic energy has a direction to it, but if you l ook closely as they turn and dip, the energy flow is pretty complicate d. To get rid of the complications and just look at the overall flow \+ we take a time average. The time average of " }{XPPEDIT 18 0 "cos(ome ga*t)^2;" "6#*$-%$cosG6#*&%&omegaG\"\"\"%\"tGF)\"\"#" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "sin(omega*t)^2;" "6#*$-%$sinG6#*&%&omegaG\"\"\"%\" tGF)\"\"#" }{TEXT -1 40 " is just 1/2 while the time average of " } {XPPEDIT 18 0 "sin(omega*t)*cos(omega*t);" "6#*&-%$sinG6#*&%&omegaG\" \"\"%\"tGF)F)-%$cosG6#*&F(F)F*F)F)" }{TEXT -1 147 " is zero. Eliminat ing the terms that average to zero and replacing the squared sines and cosines with 1/2 (I just cut and paste to do this) gives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "Savg:=-1/32*mu[0]*Io^2*BesselJ(0,k*s)*om ega*BesselY(1,k*s)*k+1/32*mu[0]*Io^2*BesselY(0,k*s)*omega*BesselJ(1,k* s)*k;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Savg:=factor(Savg) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "The Bessel function combina tion in the parentheses is a famous identity called the Wronskian. Ma ple knows how to simplify expressions containing it, like this" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Savg:=simplify(Savg,wronskian);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "So this is the simple formula for the time-averaged radiated power/area coming from the wire." } {MPLTEXT 1 0 0 "" }}}}{MARK "18" 0 }{VIEWOPTS 1 1 0 3 2 1804 }