Suppose a type of glass is such
that, for any incoming light: 70
percent of light shining from one
side is transmitted through to the
other side; 20 percent of the light is
reflected (off of the outer surface)
back in the direction from which it
came; the remaining 10 percent is
absorbed in the glass.
How much
of an original light source will be
transmitted through three panes of
glass? It is assumed that the panes
are parallel and at a small distance
from each other.
Ignore any loss
of light above or below the panes
(which is the same as assuming the
panes extend infinitely in all four
directions). Express your answer as
a ratio of integers.
Let a = 1 represent the light coming from the left onto the first pane of glass, and b represent light striking the middle pane from the left (i.e., from the direction of the first pane) and c representing the light striking the third pane coming from the same direction.
Let d be the light being directed outward (leftward as I have defined it) from the first pane, combining (as do most of these rays) both the initial reflection and any coming reflection from and transmission through the middle layer. Likewise, e is the leftward total light between the first and middle panes, and f is the total leftward light between the middle and last panes.
b=7a/10 + e/5
c=7b/10 + f/5
d=7e/10 + a/5
e=7f/10 + b/5
f=c/5
For some reason, Wolfram Alpha couldn't solve this set of simultaneous equations, so:
clearvars
c=.1;d=.1;e=.1;f=.05;
for i=1:25
b=7/10+e/5;
c=7*b/10+f/5;
d=7*e/10+1/5;
e=7*f/10+b/5;
f=c/5;
fprintf('%17.15f ',1,b,c )a
fprintf(' ')
fprintf('%17.15f ',d,e,f)
fprintf(' ')
end
sym([1 b c ])
sym([ d e f])
b=sym(b);
c=sym(c);
d=sym(d);
e=sym(e);
f=sym(f);
finds by successive approximations:
a b c
d e f
1.000000000000000 0.720000000000000 0.514000000000000
0.270000000000000 0.179000000000000 0.102800000000000
1.000000000000000 0.735800000000000 0.535620000000000
0.325300000000000 0.219120000000000 0.107124000000000
1.000000000000000 0.743824000000000 0.542101600000000
0.353384000000000 0.223751600000000 0.108420320000000
1.000000000000000 0.744750320000000 0.543009288000000
0.356626120000000 0.224844288000000 0.108601857600000
1.000000000000000 0.744968857600000 0.543198571840000
0.357391001600000 0.225015071840000 0.108639714368000
1.000000000000000 0.745003014368000 0.543230052931200
0.357510550288000 0.225048402931200 0.108646010586240
1.000000000000000 0.745009680586240 0.543235978527616
0.357533882051840 0.225054143527616 0.108647195705523
1.000000000000000 0.745010828705523 0.543237019234971
0.357537900469331 0.225055202734971 0.108647403846994
1.000000000000000 0.745011040546994 0.543237209152295
0.357538641914480 0.225055390802295 0.108647441830459
1.000000000000000 0.745011078160459 0.543237243078413
0.357538773561606 0.225055424913413 0.108647448615683
1.000000000000000 0.745011084982683 0.543237249211014
0.357538797439389 0.225055431027514 0.108647449842203
1.000000000000000 0.745011086205503 0.543237250312293
0.357538801719260 0.225055432130643 0.108647450062459
1.000000000000000 0.745011086426128 0.543237250510782
0.357538802491450 0.225055432328947 0.108647450102156
1.000000000000000 0.745011086465789 0.543237250546484
0.357538802630263 0.225055432364667 0.108647450109297
1.000000000000000 0.745011086472933 0.543237250552913
0.357538802655267 0.225055432371094 0.108647450110583
1.000000000000000 0.745011086474219 0.543237250554070
0.357538802659766 0.225055432372252 0.108647450110814
1.000000000000000 0.745011086474450 0.543237250554278
0.357538802660576 0.225055432372460 0.108647450110856
1.000000000000000 0.745011086474492 0.543237250554315
0.357538802660722 0.225055432372497 0.108647450110863
1.000000000000000 0.745011086474499 0.543237250554322
0.357538802660748 0.225055432372504 0.108647450110864
1.000000000000000 0.745011086474501 0.543237250554323
0.357538802660753 0.225055432372505 0.108647450110865
1.000000000000000 0.745011086474501 0.543237250554324
0.357538802660754 0.225055432372505 0.108647450110865
1.000000000000000 0.745011086474501 0.543237250554324
0.357538802660754 0.225055432372505 0.108647450110865
1.000000000000000 0.745011086474501 0.543237250554324
0.357538802660754 0.225055432372506 0.108647450110865
1.000000000000000 0.745011086474501 0.543237250554324
0.357538802660754 0.225055432372506 0.108647450110865
1.000000000000000 0.745011086474501 0.543237250554324
0.357538802660754 0.225055432372506 0.108647450110865
and then it finds the appropriate rational values
ans =
[1, 336/451, 245/451]
ans =
[645/1804, 203/902, 49/451]
which do indeed solve the simultaneous equations.
Since what we're going after is the amount of light exiting the last pane without reflecting back:
Since 245/451 of the original light strikes the last pane and 7/10 of that finally escapes from the triple pane set, that amount is 245*7/4510 = 343/902 of the original light shining on the first pane comes out past the last pane.
Edited on December 19, 2024, 1:46 pm
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Posted by Charlie
on 2024-12-19 13:29:27 |
computer-aided solution
