Physics for Geologists, Second edition

44 Optics
Figure 4.2 Refraction. (a) The angle i is the angle of incidence, the angle r,
the angle of refraction, and the incident medium is less dense than
the refracting medium. If you look along a partly-submerged stick
inclined at an angle i to the vertical, the submerged part appears to be
above its real position. (b) When the incident medium is denser than
the refracting medium, the angle of refraction is larger than the angle
of incidence, and there is a critical angle at which the refracted ray is
in the plane of separation of the two media (dashed line). Angles of
incidence greater than this result in total reflection. Both (a) and (b)
are incomplete because incident light is both refracted and reflected,
with two exceptions discussed in the text, at the interface between
different transparent media.
By convention, we measure the angles from the normal to the surface.
Snell's law tells us that
sin i sin R sin r
- - - ,
Ci Ci Cr
where i is the angle of incidence, R that of reflection and r that of refraction;
and ci is the speed of the incident energy or wave, and c, is the speed of
the refracted wave. In optics, the ratio ci/c, is called the refractive index,
with the symbol n, when light passes from a vacuum into a denser medium
(ci > c,). We shall pursue this shortly.
These two laws were unified by the French mathematician Pierre de Fer-
mat (1601-65) in the middle of the seventeenth century by the principle of
least time - that the light we see was the light that followed the path that
would reach our eyes in the shortest time. This implies that light travels at
different speeds in different media, slower in water than in air, for example.
In Figure 4.1 we see that pin B has its mirror-image at B1, and that the short-
est distance from B1 to A is a straight line. The shortest distance from A to B
that includes reflection from the mirror is the path that starts towards B1 and
is reflected to B. Looking at this the other way round, if a point of light were
to be placed in position B, an observer at A unable to see B directly would not
Copyright 2002 by Richard E. Chapman