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proportional to the time it wiïll take ïf the light travels with constant velocity,
1s also the sum of the bwo lengths A4 + B7. Therefore the problem becomes,
when is the sum of these two lengths the least? 'Phe answer is easy: when the
line goes through point Πas a sfraight line from A to PT In other words, we
have to fnd the point where we go toward the artificial point, and that will be
the correct one. NÑow if AC” is a straight line, then angle TP" is equal to
angle EỚP and thence to angle AC. Thus the statement that the angle of
Ineidence equals the angle of refection is equivalent to the statement that the
light goes to the mirror in such a way that it comes back to the point Ö in the
least possible time. Originally, the statement was made by Hero of Alexandria
that the light travels in such a way that it goes 0o the mirror and to the other
point in the shortest possible đZs#ønce, so 1t 1s not a modern theory. It was this
that inspired Eermat to suggest to himself that perhaps refraction operated on a
similar basis. But for refraction, light obviously does not use the path of shortest
đistance, so Fermat tried the idea that it takes the shortest #zme.
Before we go on to analyze refraction, we should make one more remark about
the mirror. IÝ we have a source of light at the point and ït sends light to
ward the mirror, then we see that the light which goes to A from the point
comes to .4 in exactly the same manner as it would have come to A ïf there were
--- Trang 458 ---
an object at , and no mirror. NÑow of course the eye detects only the light
which enters it physically, so if we have an object at and a mirror which makes
the light come into the eye in exactly the same manner as iÿ would have come
into the eye ïf the object were at ”, then the eye-brain system interprets that,
assuming i% does not know too mụch, as Öe#ng an object at 7. So the illusion
that there is an object behind the mirror is merely due to the fact that the light
which is entering the eye is entering in exactly the same manner, physically, as it
would have entered had there been an object back there (except for the dir on
the mirror, and our knowledge of the existence of the mirror, and so on, which is
corrected in the brain).
Now let us demonstrate that the principle of least time will give 5nells law
of refraction. We must, however, make an assumption about the speed of light in
water. We shall assume that the speed of light in water is lower than the speed
of light in air by a certain factor, 0ø.
`" ` E
AIR ` C
WATER = x
X À N
N ) B
Fig. 26-4. lllustration of Fermat's principle for refraction.
In Eig. 26-4, our problem is again to go from A to Ð ¡in he shortest từmne.
To illustrate that the best thíng to do is no just to go in a straight line, let us
imagine that a beautiful girl has fallen out of a boat, and she is screaming for
help in the water at point Ø. The line marked zø is the shoreline. We are at
point 4 on land, and we see the accident, and we can run and can also swim.
But we can run faster than we can swim. What do we do? Do we go in a straight
line? (Yes, no doubtl) However, by using a little more intelligence we would
realize that it would be advantageous to travel a little greater distance on land
--- Trang 459 ---
Fig. 26-5. The minimum time corresponds to point C, but nearby
points correspond to nearly the same time.
in order to decrease the distance in the water, because we øo so mụch sÌower In
the water. (Following this line of reasoning out, we would say the right thing
to do is to compu‡Èe very carefully what should be donel) At any rate, let us try
to show that the ñnal solution to the problem is the path AC, and that this
path takes the shortest time of all possible ones. Tf it is the shortest path, that
means that if we take any other, it will be longer. So, If we were to plot the time
1t takes against the position of point X, we would get a curve something like
that shown in Fig. 26-5, where point corresponds to the shortest of all possible
times. 'This means that if we move the point à to points near Œ, in the first
approximation there is essentially no change in time because the sÌope is Zero
at the bottom of the curve. So our way of ñnding the law will be to consider
that we move the place by a very small amount, and 0o demand that there be
essentially no change in tỉme. (Of course there is an infinitesimal change oŸ a
second order; we ought to have a positive increase for displacements in either
direction from Œ.) So we consider a nearby point X and we calculate how long
it would take to go from A to by the two paths, and compare the new path
with the old path. It is very easy to do. We want the diference, oŸ course, to be
nearly zero If the distance XƠ is short. Eirst, look at the path on land. lf we
draw a perpendicular à F7, we see that this path ¡is shortened by the amount EŒ.
Let us say we gain by not having to go that extra distance. Ôn the other hand,
in the water, by drawing a corresponding perpendicular, TP", we fnd that we
have to go the extra distance X”', and that is what we lose. Ôr, in #mne, we
gain the time it would have taken to go the distance EŒ, but we lose the tỉme it
would have taken to go the distance X#'. Those times must be equal since, in
the fñrst approximation, there is to be no change in time. But supposing that in
--- Trang 460 ---
the water the speed is 1/n times as fast as in air, then we must have
ĐC =n- XƑ. (26.3)
'Therefore we see that when we have the right point, XỚ sin ⁄XỨ = n- XC sin XŒT"
or, cancelling the common hypotenuse length XΠand noting that
EXC = ECN =0, and XCF~ BƠN ' =0, (when X is near C),
we have
sin Ø; = nsin Ø„. (26.4)
So we see that to get from one point to another in the least time when the ratio
of speeds is nø, the light should enter at such an angle that the ratio of the sines
of the angles Ø, and đ, is the ratio of the speeds in the two media.
26-4 Applications of Fermat”s principle
Now let us consider some of the interesting consequences of the principle of
least tìme. First is the prineiple of reciprocity. Ifto go from 4 to we have found
the path of the least time, then to go in the opposite direction (assuming that
light goes at the same speed in any direction), the shortest time will be the same
path, and therefore, If light can be sent one way, it can be sent the other way.
An example of interest is a glass block with plane parallel faces, set at an
angle to a light beam. Light, in going through the block from a point A to a
point Ö (Eig. 26-6) does not go throuph in a straight line, but instead it decreases
the time in the block by making the angle in the bloeck less inclined, although it
loses a little bit in the aïir. The beam is simply displaced parallel to itself because
the angles in and out are the same.
A third interesting phenomenon is the fact that when we see the sun setting,