Gem-Stones and Their Distinctive Characters

by George Frederick Herbert Smith

Ordinary experience with a plane mirror tells us that, when light is returned, or reflected, as it is usually termed, from a plane or flat surface, there is no alteration in the size of objects viewed in this way, but that the right and the left hands are interchanged: our right hand becomes the left hand in our reflection in the mirror. We notice, further, that our reflection is apparently just as far distant from the mirror on the farther side as we are on this side. In Fig. 13 MM´ is a section of the mirror, and O´ is the image of the hand O as seen in the mirror. Light from O reaches the eye E by way of m, but it appears to come from O´. Since OO´ is perpendicular to the mirror, and O and O´ lie at equal distances from it, it follows from elementary geometry that the angle i´, which the reflected ray makes with mn, the normal to the mirror, is equal to i, the angle which the incident ray makes with the same direction.

Fig. 13.—Reflection at a Plane Mirror.

Again, everyday experience tells us that the case is less simple when light actually crosses the bounding surface and passes into the other medium. Thus, if we look down into a bath filled with water, the bottom of the bath appears to have been raised up, and a stick plunged into the water seems to be bent just at the surface, except in the particular case when it is perfectly upright. Since the stick itself has not been bent, light evidently suffers some change in direction as it passes into the water or emerges therefrom. The passage of light from one medium to another was studied by Snell in the seventeenth century, and he enunciated the following laws:—

1. The refracted ray lies in the plane containing the incident ray and the normal to the plane surface separating the two media.

It will be noticed that the reflected ray obeys this law also.

2. The angle r, which the refracted ray makes with the normal, is related to the angle i, which the incident ray makes with the same direction, by the equation

n sin i = n´ sin r, (a)

where n and n´ are constants for the two media which are known as the indices of refraction, or the refractive indices.

This simple trigonometrical relation may be expressed in geometrical language. Suppose we cut a plane section through the two media at right angles to the bounding plane, which then appears as a straight line, SOS´ (Fig. 14), and suppose that IO represents the direction of the incident ray; then Snell’s first law tells us that the refracted ray OR will also lie in this plane. Draw the normal NON´, and with centre O and any radius describe a circle intersecting the incident and refracted rays in the points a and b respectively; let drop perpendiculars ac and bd on to the normal NON´. Then we have n.ac = n´.bd, whence we see that if n be greater than n´, ac is less than bd, and therefore when light passes from one medium into another which is less optically dense, in its passage across the boundary it is bent, or refracted, away from the normal.