Page:Elementary Text-book of Physics (Anthony, 1897).djvu/421

§334] If we represent the angle of incidence $$moN$$ by $$i,$$ and the angle of refraction $$poN'$$ by $$r,$$ we have This constant is called the index of refraction. This is the expression of Snell's law of refraction. Here again the time required for the light to pass by $$mop$$ from $$m$$ in one medium to $$p$$ in the other is less than by any other path.

We may now trace a wave through a medium bounded by plane surfaces. Suppose the wave 'front' and bounding planes of the medium all perpendicular to the plane of the paper. We shall have as above for the first surface $$\frac{\sin i}{\sin r} = \frac{v}{v'} = \mu,$$ and for the second surface $$\frac{\sin i'}{\sin r'} = \frac{v'}{v''} = \mu '.$$

If, as is often the case, the light emerge into the first medium, $$v'' = v,$$ and $$\frac{\sin i'}{\sin r'} = \frac{v'}{v} = \frac{1}{\mu}\cdot$$

If the bounding planes be parallel, $$i' = r,$$ and we have $$\frac{\sin r}{\sin r'} = \frac{1}{\mu};$$ hence $$i = r',$$ or the incident and emergent waves are parallel. If the two bounding planes form an angle $$A$$ the body is called a prism. The wave incident upon the second face will make with it an angle $$A - r,$$ and the emergent wave is found by the relation $$\frac{\sin (A - r)}{\sin r'} = \frac{1}{\mu}$$ or $$\frac{\sin r'}{\sin (A - r)} = \mu.$$ The direction of the emerging wave front may be found by construction.

Draw $$Ai$$ (Fig. 100) parallel to the incident wave. From some point $$B$$ on $$AB$$ describe an arc tangent to $$Ai;$$ from the same point with a radius $$\frac{Bi}{\mu}$$ describe the arc $$rr.$$ $$Ar,$$ tangent to $$rr,$$ is the refracted wave front. From some point $$C$$ on