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it is already below the horizonl It does not iook as though it is below the horizon,
but it is (Eig. 26-7). The earth's atmosphere is thin at the top and dense at the
bottom. Light travels more slowly in air than it does in a vacuum, and so the
light of the sun can get to point Š beyond the horizon more quickly if, instead
of Jjust going in a straight line, it avoids the dense regions where I% goes sÌowly
by getting through them at a steeper tilt. When it appears to go below the
horizon, it is actually already well below the horizon. Another example of this
phenomenon is the mirage that one often sees while driving on hot roads. Ône
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—=————— ——-----S
<3 _ l
Fig. 26-6. A beam of light is offset as It passes through a transparent
block.
TO APPARENT SUN
P << LIGHT PATH
⁄ TO TRUE
EARTH
Fig. 26-7. Near the horizon, the apparent sun ¡is higher than the true
sun by about 1/2 degree.
HOT ROAD OR SAND
Fig. 26-8. A mirage.
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sees “water” on the road, but when he gets there, it is as dry as the desertl The
phenomenon is the following. What we are really seeing is the sky light “reflected”
on the road: light from the sky, heading for the road, can end up in the eye, as
shown in Fig. 26-8. Why? The aïr is very hot just above the road but it is cooler
up higher. Hotter air is more expanded than cooler air and is thinner, and this
decreases the speed of light less. 'Phat is to say, light goes faster in the hot region
than in the cool region. Thherefore, instead of the light deciding to come in the
straightforward way, it also has a least-time path by which it goes into the region
where it goes faster for awhile, in order to save time. So, iÿ can øo in a curve.
QIi„~” |
P I I P'
¡ OPTICAL SYSTEM ¡
Fig. 26-9. An optical “black box.”
As another important example of the principle of least time, suppose that
we would like 0o arrange a situation where we have all the light that comes out
of one point, , collected back together at another point, P7 (Eig. 26-9). That
means, oŸ course, that the light can go in a straight line from to P'. That is all
ripht. But how can we arrange that not only does it go straight, but also so that
the light starting out from toward @ also ends up at P“? We want to bring all
the light back to what we call a ƒocus. How? T the light always takes the path of
least time, then certainly it should not want to go over all these other paths. The
only way that the light can be perfectly satisfed to take several adjacent paths
1s to make those times ezøcf equal Otherwise, it would select the one of least
time. “Therefore the problem of making a focusing system is merely to arrange a
device so that it takes the same time for the light to go on ai the diferent pathsl
'This is easy to do. Suppose that we had a piece of glass in which light goes
slower than it does in the air (Fig. 26-10). Now consider a ray which goes ïn air in
the path PQP. That is a longer path than from directly to P7“ and no doubt
takes a longer time. But if we were to insert a piece of glass of Just the right
thickness (we shall later ñgure out how thick) it might exactly compensate the
excess time that it would take the light to go at an anglel In those circumstances
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Fig. 26-10. A focusing optical system.
we can arrange that the time the light takes to go straight through is the same
as the time it takes to go ïn the path PQP”. Likewise, if we take a ray PRINP/
which is partly inclined, i% is not quite as long as PỌQP”, and we do not have
to compensate as mụuch as for the straight one, but we do have to compensate
somewhat. We end up with a piece of glass that looks like Eig. 26-10. With this
shape, all the light which comes from will go to PP, Thịis, of course, is well
known to us, and we call such a device a converging /ens. In the next chapter we
shall actually calculate what shape the lens has to have to make a perfect focus.
Fig. 26-11. An ellipsoidal mirror.
'Take another example: suppose we wish to arrange some mirrors so that the
light from ? always goes to P' (Eig. 26-11). Ôn any path, it goes to some mirror
and comes back, and all times must be equal. Here the light always travels in
air, so the time and the distance are proportional. Therefore the statement that
all the times are the same is the same as the statement that the total distance 1s
the same. Thus the sum of the two distances r¡ and 7s must be a constant. An
cllipse 1s that curve which has the property that the sum of the distances from
two poinfs is a constant for every point on the ellipse; thus we can be sure that
the light from one focus will come to the other.
The same principle works for gathering the light of a star. The great 200-inch
Palomar telescope is built on the following principle. Imagine a star billions of
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K II cIP ⁄
SEIIR Ị
N\ | =<=1⁄
"x. >.^
có CIỦI ¡
I I Mà. I I
Fig. 26-12. A paraboloidal mirror.
miles away; we would like to cause all the light that comes in to come to a Íocus.
Of course we cannot draw the rays that go all the way up to the star, but we
still want to check whether the times are equal. Of course we know that when
the various rays have arrived at some plane ##“”, perpendicular to the rays, all
the tỉimes in this plane are equal (Eig. 26-12). The rays must then come down
to the mirror and proceed toward ¡n equal times. That is, we must fnd a
curve which has the property that the sum of the distances XX” + X”P' is a
constant, no matter where X is chosen. An easy way to find it is to extend the
length of the line XX” down to a plane ÙE. Now 1ƒ we arrange our curve so that
.A“=AP, BE" =BT', CC” = C”P', and so on, we will have our curve,
because then of course, 4A” + AP' = AA' + A7A” will be constant. Thus our
curve is the locus of all points equidistant from a line and a point. Such a curve
1s called a parabola; the mirror is made in the shape of a parabola.
'The above examples illustrate the principle upon which such optical devices
can be designed. The exact curves can be calculated using the principle that, to
focus perfectly, the travel times must be exactly equal for all light rays, as well
as being less than for any other nearby path.