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{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" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "LENGTH:  Lecture-lab demo
nstration." }}{PARA 0 "" 0 "" {TEXT -1 47 "AUDIENCE:  Student use in c
omputer lab setting." }}{PARA 0 "" 0 "" {TEXT -1 21 "SOFTWARE: Maple V
 R 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "F
UTURE:  Add a diagram of refraction to supplement the discussion." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "HISTORY: \+
 Created December 3, 1997" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 13 "REFERENCES:  " }{TEXT 258 22 "Fundamentals of Opti
cs" }{TEXT -1 76 ", 2nd Edition; by F. A. Jenkins, H. E. White, McGraw
-Hill, 1950, pages 6-8. " }}{PARA 0 "" 0 "" {TEXT 259 31 "Geometrical \+
and Physical Optics" }{TEXT -1 38 ", 2nd ed.  Longhurst, 1967, pages 6
-9." }}{PARA 0 "" 0 "" {TEXT 257 16 "Calculus, 2nd ed" }{TEXT -1 75 ".
  by Hughes-Hallett, Gleason, et al. J. Wiley, 1997, page 277, Problem
 22." }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 33 "Refraction and Fermat's P
rinciple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 11 "Definitions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 139 "When a ray of light travels from one med
ium to another (for example, from air to water) it changes direction. \+
 This phenomenon is known as " }{TEXT 256 10 "refraction" }{TEXT -1 
90 ".  The amount of refraction depends on the velocities v1 and v2 of
 light in the two media." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The refra
ctive index " }{TEXT 261 1 "n" }{TEXT -1 98 " of a medium is the ratio
 of the speed of light in the medium to the speed of light in a vacuum
:  " }{TEXT 262 19 "n = V/c, so V = nc." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 30 "When light travels a distance " }
{TEXT 263 1 "d" }{TEXT -1 33 " in a medium of refractive index " }
{TEXT 264 1 "n" }{TEXT -1 33 " the optical path is the product " }
{TEXT 265 3 "nd." }}{PARA 0 "" 0 "" {TEXT -1 220 "Thus the optical pat
h is the distance the light travels in the medium in the same time tha
t light would travel d in a vacuum.  Optical path is a measure of time
 of travel in a medium, but is more general than just time. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 260 18 "Fermat's Principle" }
{TEXT -1 172 " states that the path along which light travels is such \+
that the total optical path \005taken is at a minimum or maximum (also
 called a critical value, or a stationary value.)" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 295 "Notation and Set
-Up:  From a point A at a distance a above the interface, lights trave
ls to point B below the interface.  See the diagram.  The refractive i
ndex is n1 in the upper medium (containing starting point A).  The ref
ractive index is n2 in the lower medium (containing ending point B). \+
" }}{PARA 0 "" 0 "" {TEXT -1 45 "The horizontal distance between A and
 B is c." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
104 "Let the point R where the light ray crosses the interface between
 the media be at x horizontally from B." }}{PARA 0 "" 0 "" {TEXT -1 
71 "The question is how to use Fermat's Principle to choose the distan
ce x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 10 "Derivation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 33 "Express the distance from A to R." }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "dAR := ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 37 "Express the optical path from A to R." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "opAR := ;
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
33 "Express the distance from R to B:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "dRB := ;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "\005Expre
ss the optical path from R to B" }}{PARA 0 "" 0 "" {TEXT -1 1 ":" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "opRB := ;" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Now express the t
otal optical path:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 12 "totalop := ;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Find the derivative of th
e total optical path with respect to the position x." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Dtotalop \+
:= ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fermat_eqn := Dtota
lop = 0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 105 "Imagine that the equation is solved for, and the critica
l point is x0.  Use this in the Fermat Principle:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "new_fermat_
eqn := subs( x=x0, fermat_eqn);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "readlib(isolate);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "isolate(new_fermat_eqn, n1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "\"/n2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s
implify(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}
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