When a ray of light falls from a denser upon a rarer medium, as in passing from water to air, it will suffer such a change in its direction, that the sines of the angles of incidence and refraction will be in a constant ratio.
Let EC, Fig. 1. be a ray which is incident at C, on the rarer medium above RS, then if EC be the refracted ray, when AC is the incident ray from the rarer medium, it is found that AC is the refracted ray when EC is the incident ray.
This is obvious, from the simple experiment men tioned under the last proposition; for since the points A, C, B, are seen in a straight line, it follows that the rays of the object at E reach the eye along the line CA. Hence it follows, that the sines of the angles of incidence and refraction are in the same ratio as when AC was the incident ray ; that is, in a constant ratio.
When the refraction is made from water into air, this ratio is 4 to 3, or as 1.33 to 1, and so on Nvith other bodies.
COR. When a ray passes from a denser into a rarer medium, the refraction is always made from the per pendicular.
The general truth of this corollary is well shoNvn by the common experiment known since the time of Ptole my, of rendering a shilling at the bottom of a vessel vi sible by refraction. Jr AVC place a shilling at the point E, Fig. 1. and suppose CQES to be the opaque sides of a vessel, then it is clear that the eye placed at A will not see the shilling at E, if there is no water in the vessel CQES. The morncnt, however, that it is filled with water, the ray EC, issuing from the shilling, will be bent from the perpendicular CP, in the direction CA, so as to leach the eye of the observer at A.
If a ray of light is incident from a denser upon a rarer medium, it is not capable of being refracted, but will return into the rarer medium, when the angle of inci dence is greater than that at which the Sine of the angle of refraction becomes equal to the Radius.
Let AC, Fig. 2. be the ray incident upon the rarer medium RS. It Nvill be refracted from the perpendicu lar DF into the direction CE, so that AD is to EF in a constant ratio. If we increase the angle ACD, the angle FCE will also increase till the lines CE and FE coin cide with the radius CS. But if beyond this position of the ray AC, the angle ACD is still farther increased, it is manifest that its Sine is also increased ; and, conse quently, in order that the ratio may be constant, the Sine of refraction EF must also increase, which is impossible, as it is already by hypothesis equal to the radius CS.
Hence it follows, that whenever the angle of incidence is greater than that at which the Sine of the angle of fraction becomes equal to the radius, the ray cannot be refracted consistently with the constant ratio of the sines.
This is found to be the case by experiment ; and at the angle thus indicated, all the incident rays arc re flected from the inner surface of the denser medium, having a reflexion more brilliant than what can be pro duced from any metallic surtace. The reflexion is then called Total Rtile.rion.
The angle at which total reflexion takes place may be thus found : Lct x be the sine of incidence at which the corresponding sine of refraction is 1, and let vz. represent the index of refraction, or the ratio or the sincs; then x : 1 = 1: 2/Z by Prop. and x=— 1, that is, total reflcxion takes place when the sine of in cidence is equal to the reciprocal of the index of re fraction.
In Water, whose index of refraction is 1 336, the an gle of total reflexion is 48' 28' In Glass, whose index of refraction is 1.50, it is 41° 49'.
In Zircon and Sulphur, whose index of refraction is about 2.00, it is 30° 0'.
In Diamond, whose index of refraction is about 2,50, it is 23° 35'.
If ni and mi are the indices of refraction from a va cuum into any two media of different refractive powers, the index of refraction for a ray of light passing out or the first or the rarest, into the second or the densest, will be —mf.
7/1 Let a ray of light be incident from air upon a parallel plate of water, whose index of refraction is 1/1, lying upon a parallel plate of glass, whose index is m', then it is manifest that the angle of incidence from the water upon the glass, is the angle of refraction from the air into the water, and consequently their sines are to the sine of incidence as 1: nz. In like manner, the angle of refraction from the water into the glass, is the angle of incidence from the glass into the air, and quently its sine is to that of the angle of refraction, from glass into air, as 1 : 772'. Hence the since of dence trona water to glass is to the sine of refraction, : 272, Or to make one or these unity, we have 722' 771 : 7n' = I : In Water aud Wass, when m and 7/1' arc 1.525 and m' 1,336, the index of refraction is —= 1.141.