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We shall discuss these focusing optical devices further in the next chapter; let
us now discuss the further development of the theory. When a new theoretical
principle ¡is developed, such as the principle of least time, our first inclination
might be to say, “Well, that is very pretty; it is delightful; but the question is,
does it help at all in understanding the physics?” Someone may say, “Yes, look
at how many things we can now understand!” Another says, “Very well, but ÏI
can understand mirrors, too. Ï need a curve such that every tangent plane makes
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equal angles with the bwo rays. I can figure out a lens, too, because every ray
that comes to it is bent through an angle given by Snells law.” Evidently the
statement of least time and the statement that angles are equal on refection,
and that the sines of the angles are proportional on refraction, are the same. So
1s Iÿ merely a philosophical question, or one of beauty? 'There can be arguments
on both sides.
However, the Importance of a powerful prineiple is that #‡ predicts neu things.
Tt is easy to show that there are a number oŸ new things predicted by Fermaf”s
principle. First, suppose that there are f£hree media, glass, water, and aïr, and
we perform a refraction experiment and measure the index ø for one medium
against another. Let us call ma the index of air (1) against water (2); ma the
index of air (1) against glass (3). IÝ we measured water against glass, we should
fnd another index, which we shall call nmạs. But there is no ø pr?or¿ reason why
there should be any connection between 01s, 01s, and 2s. Ôn the other hand,
according to the idea of least time, there 2s a defnite relationship. The index ?1a
1s the ratio of two things, the speed in air to the speed in water; 1s is the ratio
of the speed ín air to the speed in gÌlass; 23 is the ratio of the speed in water to
the speed in glass. 'herefore we cancel out the air, and get
¬......` (26.5)
U3 ĐỊ (0a T2
In other words, we ørcd¡¿ct that the Index for a new pair of materials can be
obtained from the indexes of the individual materials, both against air or against
vacuum. 5o iŸ we measure the speed of light in all materials, and from this get a
single number for each material, namely its index relative to vacuum, called m¿
(mị is the speed in air relative to the speed in vacuum, etc.), then our formula is
easy. The index for any two materials 2 and 7 is
ng= TƯ = 2, (26.6)
Uj Tt¿
Using only Snells law, there is no basis for a prediction of this kind.* But of
course this prediction works. The relation (26.5) was known very early, and was
a very strong argument for the prineciple of least time.
— * Although it can be deduced if the additional assumption is made that adding a layer of
one substance to the surface of another does not change the eventual angle of refraction in the
latter material.
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Another argument for the principle of least time, another prediction, is that
1 we measure the speed of light in water, it will be lower than in air. This is a
prediction of a completely diferent type. It is a brilliant prediction, because all
we have so far measured are øngles; here we have a theoretical prediction which
1s quite diferent from the observations from which Eermat deduced the idea of
least time. It turns out, in fact, that the speed in water 2s slower than the speed
in air, by just the proportion that is needed to get the right indexl
26-5 A more precise statement of Fermat?s principle
Actually, we must make the statement of the principle of least time a little
more accurately. It was not stated correctly above. It is #øcorrectu called the
principle of least time and we have gone along with the incorrect description Íor
convenience, but we must now see what the correct statement is. Suppose we had
a mirror as in Fig. 26-3. What makes the light think it has to go to the mirror?
The path of ieast time is clearly 4P. So some people might say, “Sometimes it is
a maximum time.” Ít is no‡ a maximum time, because certainly a curved path
would take a stil longer timel The correct statement is the following: a ray going
in a certain particular path has the property that if we make a small change (say
a one percent shift) in the ray in any manner whatever, say in the location at
which it comes to the mirror, or the shape of the curve, or anything, there will
be mo first-order change in the time; there will be only a second-order change in
the time. In other words, the principle is that light takes a path such that there
are many other paths nearby which take almost exactly the sazne tỉme.
The following is another difculty with the principle of least time, and one
which people who do not like this kind of a theory could never stomach. With
Snells theory we can “understand” light. Light goes along, 1% sees a surface, 1%
bends because it does something at the surface. “The idea of causality, that it goes
from one point to another, and another, and so on, is easy to understand. But
the prineiple of least time is a completely diferent philosophical principle about
the way nature works. Instead oŸ saying it is a causal thing, that when we do one
thing, something else happens, and so on, it says this: we set up the situation,
and igh# decides which is the shortest time, or the extreme one, and chooses
that path. But uhø‡ does it do, ho does it nd out? Does 1t srneÏl the nearby
paths, and check them against each other? The answer is, yes, it does, in a way.
That is the feature which is, of course, not known in geometrical optics, and
which is involved ïn the idea of auelength; the wavelength tells us approximately
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—T^ \ |) |_— :91) --H[D]
Fig. 26-13. The passage of radiowaves through a narrow silit.
how far away the light must “smell” the path in order to check it. Tt is hard
to demonstrate this fact on a large scale with light, because the wavelengths
are so t6erribly short. But with radiowaves, say 3-cm waves, the distances over
which the radiowaves are checking are larger. lf we have a source of radiowawves,
a detector, and a slit, as in Eig. 26-13, the rays of course go from ,Š to J because
1t is a straight line, and if we close down the slit it is all right—they still go. But
now if we move the detector aside to J2, the waves will not go through the wide
slit from 6 to D/, because they check several paths nearby, and say, “No, my
friend, those all eorrespond to diferent times.” Ôn the other hand, if we preuen‡
the radiation from checking the paths by closing the slit down to a very narrow
crack, then there is but one path available, and the radiation takes it! With a
narrow slit, more radiation reaches than reaches it with a wide slitl
One can do the same thing with light, but it is hard to demonstrate on a large
scale. The efect can be seen under the following simple conditions. Eind a small,
bright light, say an unfrosted bulb in a street light far away or the reflection of
the sun in a curved automobile bumper. Then put two fingers in front of one eye,
so as to look through the crack, and squeeze the light to zero very gently. You
will see that the image of the light, which was a little dot before, becomes quite
elongated, and even stretches into a long line. 'Phe reason is that the ñngers are