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Now the curvatures of the arcs co,d and co,d are measured by the reciprocals of their respective radii*; that is,
Curvature of co,d
Now when the arcs are small, a condition which in general is realized in experimental work, their curvatures are proportional to the extent to which they bulge out from the straight line cod; that is,
From (1) and (2) we get
Speed in air
Speed in water
speed in air
speed in water
*Construct an angle of 45° (Fig. 393, angle formed by the tangents t, t' is 45°. great (Fig. 393, (2)) draw an arc whose length is equal to that of the arc in Fig. 393 (1). Since the radius is three times as great, this arc contains 15°, and the angle formed by the tangents is 15°. From this we see that the arc whose radius is three times as great curves, or changes its direction, one third as fast; that is, the change in curvature of an arc of given length varies inversely with the radius. In general then, the curvature of an arc is measured by the reciprocal of its radius.
† oc (Fig. 394) is a mean proportional between the two segments of the diameter; hence ao x od oc2. For very small arcs od is practically equal to the diameter 2r. Hence 062 Oc2 1
- or ao = 21
X-• Therefore ao is proportional to
That is. the distances to which two small arcs having a
(1)). Its are contains 45° and the Now with a radius three times as
common chord bulge out from the chord are proportional to the respective curvatures of the arcs.
But in looking vertically downward, as in the experiment
with the jar of water,
Speed in air
Speed in water OP'
But in our experiment we found
oP real depth
that the bottom was apparP 4 ently raised one fourth of the depth; that is, that OP 3 We conclude, therefore, that light travels three fourths as fast in water as in air.
The fact that the value of this ratio, as determined by this indirect method, is exactly the same as that found by Foucault and Michelson (see opposite p. 358) by direct measurement (§ 419) furnishes one of the strongest proofs of the correctness of the wave theory.
432. Index of refraction. The ratio of the speed of light in air to its speed in any other medium is called the index of refrac tion of that medium. It is evident that the method employed. in the last paragraph for determining the index of refraction of water can be easily applied to any transparent medium whether liquid or solid.* The refractive indices of some of the commoner substances are as follows:
433. Light waves are transverse. Thus far we have discovered but two differences between light waves and sound waves; namely, the former are disturbances in the ether and are of very short wave length, while the latter are disturbances in
*To show the extreme beauty, simplicity, and accuracy of this method of getting index of refraction it is suggested that the teacher use the following method in his laboratory work.
A very sharp pencil must be used for this exercise. Make a dot P on a sheet of paper. Put the glass plate (Fig. 395, (1)) on the sheet so that the
ordinary matter and are of relatively great wave length. There exists, however, a further radical difference which follows from a capital discovery made by Huygens (see opposite p. 364) in the year 1690. It is this: While sound waves consist, as we have already seen, of longitudinal vibrations of the particles of the transmitting medium, that is, vibrations back and forth in the line of propagation of the wave, light waves are like the water waves of Fig. 346, p. 324, in that they consist of transverse vibrations, that is, vibrations of the medium at right angles to the direction of the line of propagation.
In order to appreciate the difference between the behavior of waves of these two types under certain conditions, conceive
edge of the label pasted around the edge of the glass coincides with the dot (or in case a prism (Fig. 395, (2)) is used, let the apex P coincide with the dot). Draw the base line ef and the other sides of the glass, holding it firmly down meanwhile. Be sure that at no time during the exercise does the glass slip the slightest from its first position. Lay a ruler upon the paper in a slantwise position cd (not touching the glass), and, with one eye closed, make its edge point ex
actly at the apparent position of P as seen through the glass. If you are now sure that your ruler did not push the glass out of position, draw a line cd with the
FIG. 395. Index of refraction
sharp pencil. Similarly, draw another line ab about as far to the right of the center as cd is to the left. Remove your glass and complete the drawing as indicated in the diagram.
P' is the apparent position of P. As you have already learned from your text, the ratio of the velocities of light in air and glass is found by dividing dP by dP'. Measure these distances very carefully to 0.1 mm., and calculate the index of refraction to two decimal places. Make two or three more trials and compare results.
of transverse waves in a rope being made to pass through two gratings in succession, as in Fig. 396. So long as the slits in both gratings are parallel to the plane of vibration of the hand, as in Fig. 396, (1), the waves can pass through them with perfect ease; but if the slits in the first grating Pare parallel to the direction of vibration, while those of the second grating Q are turned at right angles to this direction, as in Fig. 396, (2), it is evident that the waves will pass readily through P, but will be stopped completely by Q, as shown in the figure. In other words, these gratings P and Q will let through only such vibrations as are parallel to the direction of their slits.
FIG. 396. Transverse waves passing
If, on the other hand, a longitudinal instead of a transverse wave-such, for example, as a sound wave had approached such a grating, it would have been as much transmitted in one position of the grating as in another, since a to-and-fro motion of the particles can evidently pass through the slits with exactly the same ease, no matter how they are turned.
Now two crystals of tourmaline are found to behave with respect to light waves just as the two gratings behave with respect to the waves on the rope.
FIG. 397. Tourmaline tongs
Let one such crystal a (Fig. 397) be held in front of a small hole in a screen through which a beam of sunlight is passing to a neighboring wall; or, if the sun is not shining, simply let the crystal be held between the eye and a source of light. The light will be readily transmitted, although somewhat diminished in intensity. Then let a second crystal b be held in line with the first. The light will still be transmitted, provided the axes of
the crystals are parallel, as shown in Fig. 398. When, however, one of the crystals is rotated in its ring through 90° (Fig. 399), the light is cut off.
This shows that a crystal of tourmaline is capable of transmitting only light which is vibrating in one particular plane.
FIG. 398. Light pass-
FIG. 399. Light cut off by crossed tourmaline crystals
From this experiment, therefore, we are forced to conclude that light waves are transverse rather than longitudinal vibrations. The experiment illustrates what is technically known as the polarization of light, and the beam which, after passage through a, is unable to pass through b if the axes of a and b are crossed, is known as a polarized beam. It is, then, the phenomenon of the polarization of light upon which we base the conclusion that light
waves are transverse.
434. Intensity of illumination. Let four candles be set as close together as possible in such a position B as to cast upon a white screen C', placed in a well-darkened room, a shadow of an opaque object O (Fig. 400). Let one single candle be placed in a position A such that it will cast another shadow of O upon the screen. Since light from A falls on the shadow cast by B, and light from B falls on the shadow cast by A, it is clear that the two shadows will appear equally dark only when light of equal intensity falls on each; that is, when A and B produce equal illumination upon the screen. Let the positions of A and B be shifted until this condition is fulfilled. Then let the distances
FIG. 400. Rumford's photometer
from B to C and from A to C be measured. If all five candles are burning with flames of the same size, the first distance will be found to be just twice as great as the second. Hence the illumination produced upon the screen by each one of the candles at B is but one fourth as great as that produced on the screen by one candle at A, one half as far away.