{VERSION 2 3 "IBM INTEL NT" "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 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Fo nt 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 255 0 0 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 2 2 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 80 "PHYSICS RESOURCE PACKETS P ROJECT File: LICENSE.MWS" }}{PARA 0 "" 0 "" {TEXT -1 110 "Rose-Hulman Institute of Technology \+ Authors: Perry Peters & Greg Williby" }}{PARA 0 "" 0 "" {TEXT -1 97 "http://www.rose-hulman.edu/~moloney \+ Software: Maple V Release 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 404 "This problem has light rays coming parallel to the axis and re fracting inside a sphere then coming back out. The idea is to show tha t for a range of index of refraction one will get rays coming back out which are mostly parallel to the axis, so making a spherical bead a g ood approximation to a retro-reflector. Small beads of this type are s old in 3M tape to go on license plates and other applications." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "Perry sh ows n=1.85 below, but one would want to have a way to compare n=1.85 t o other indices of refraction for effectiveness." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Index of refraction." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "n:=1.85;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "i:=0.05;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 29 "Position of horizontal lines." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "for i from 0.05 to 1 by 0.05 do" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Equation of incoming ray. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "L1[i]:=i;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Find where incoming ray h its the circle." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "sol[i]:=fsolve( \{y1[i]=L1[i],x1[i]=cos(theta1[i]),y1[i]=sin(theta1[i])\},\{theta1[i], x1[i],y1[i]\},\{theta1[i]=-Pi/2..Pi/2\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assign(sol[i]);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "Find the angle of the refracted ray." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "snell[i]:=1*sin(theta1[i])=n*sin(th eta2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "theta2[i]:=solve(snell[i] ,theta2);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Find the slope angle of the refracted ray." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "thetaL2[i]:=theta1[i]-theta2[i];" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Equation of the line for \+ the refracted ray." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "L2[i]:=tan(th etaL2[i])*(x-x1[i])+y1[i];" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Find where the refracted ray intersects the circl e." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "sol[i]:=fsolve(\{y2[i]=subs( x=x2[i],L2[i]),x2[i]=cos(theta3[i]),y2[i]=sin(theta3[i])\},\{theta3[i] ,x2[i],y2[i]\},\{theta3[i]=Pi/2..3*Pi/2\}); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assign(sol[i]);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "Find the slope angle of the reflected ray ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "thetaL3[i]:=thetaL2[i]-2*theta 2[i];" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 " Equation of the line for the reflected ray." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "L3[i]:=tan(thetaL3[i])*(x-x2[i])+y2[i];" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Find where the ref lected ray intersects the circle." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "sol[i]:=fsolve(\{y3[i]=subs(x=x3[i],L3[i]),x3[i]=cos(theta4[i]),y 3[i]=sin(theta4[i])\},\{theta4[i],x3[i],y3[i]\},\{theta4[i]=Pi..2*Pi\} ); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "assign(sol[i]);" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Find the slope ang le of the reflected ray leaving the circle." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "thetaL4[i]:=theta4[i]+theta1[i];" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Equation of the line for \+ the ray leaving the circle." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "L4[i ]:=tan(thetaL4[i])*(x-x3[i])+y3[i];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Make plots for the inco mming and outgoing rays." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "for i f rom 0.05 to 1 by 0.05 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "f.(tru nc(i*20)):=plot(\{[x,L1[i],x=x1[i]..2],[x,L2[i],x=x2[i]..x1[i]]\},colo r=blue,scaling=constrained):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "h. (trunc(i*20)):=plot(\{[x,L3[i],x=x2[i]..x3[i]],[x,L4[i],x=x3[i]..2]\}, color=green,scaling=constrained):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Make a plot of the circle ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "g:=plot([cos(theta),sin(theta) ,theta=0..2*Pi],color=black):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 " Display all the plots on the same graph." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "display(\{g,f.(1..20),h.(1..20)\});" }}}}{MARK "7 0 0" 30 } {VIEWOPTS 1 1 0 1 1 1803 }