To Explain the World: The Discovery of Modern Science (59 page)

where if arcsin
x
is measured in degrees, then
R
= 360°/2
π
. Thus when the angle of incidence varies by an amount
δi
, the angle of deflection changes by

or, since
δ
sin
i
= cos
i δi
/
R
,

Hence the condition for a maximum of
φ
is that

Squaring both sides of the equation and using cos
²
i
= 1 – sin
²
i
(which follows from the theorem of Pythagoras), we can then solve for sin
i
, and find

At this angle,
φ
takes its maximum value:

For
n
= 4/3, the maximum value of
φ
is reached for
b
/
R
= sin
i
= 0.86, for which
i
= 59.4°, where
r
= 40.2°, and
φ
_max
= 42.0°.

30. Wave Theory Derivation of the Law of Refraction

The law of refraction, which as described in Technical Note 28 can be derived from an assumption that refracted light rays take the path of least time, can also be derived on the basis of the wave theory of light. According to Huygens, light is a disturbance in a medium, which may be some transparent material or space that is apparently empty. The front of the disturbance is a line, which moves forward in a direction at right angles to the front, at a speed characteristic of the medium.

Figure 23. Refraction of a light wave.
The horizontal line again marks the interface between two transparent media, in which light has different speeds. The crosshatched lines show a segment of a wave front at two different times—when the leading edge and when the trailing edge of the wave front just touch the interface. The solid lines marked with arrows show the paths taken by the leading and trailing edges of the wave front.

Consider a segment of the front of such a disturbance, which is of length
L
in medium 1, traveling toward an interface with medium 2. Let us suppose that the direction of motion of the disturbance, which is at right angles to this front, makes angle
i
with the perpendicular to this interface. When the leading edge of the disturbance strikes the interface at point
A
, the trailing edge
B
is still at a distance (along the direction the disturbance is traveling) equal to
L
tan
i.
(See Figure 23.) Hence the time required for the trailing edge to reach the interface at point
D
is
L
tan
i
/
v
₁
, where
v
₁
is the velocity of the disturbance in medium 1. During this time the leading edge of the front will have traveled in medium 2 at an angle
r
to the perpendicular, reaching a point
C
at a distance
v
₂
L
tan
i
/
v
₁
from A, where
v
₂
is the velocity in medium 2. At this time the wave front, which is at right angles to the direction of motion in medium 2, extends from
C
to
D
, so that the triangle with vertices
A
,
C
, and
D
is a right triangle, with a 90° angle at
C.
The distance
v
₂
L
tan
i
/
v
₁
from
A
to
C
is the side opposite angle
r
in this right triangle, while the hypotenuse is the line from
A
to
D
, which has length
L
/cos
i.
(Again, see Figure 23.) Hence

Recalling that tan
i
= sin
i
/cos
i
, we see that the factors of cos
i
and
L
cancel, so that

sin
r
=
v
₂
sin
i
/
v
₁

or in other words

which is the correct law of refraction.

It is not an accident that the wave theory, as worked out by Huygens, gives the same results for refraction as the least-time principle of Fermat. It can be shown that, even for waves passing through a heterogeneous medium in which the speed of light changes gradually in various directions, not just suddenly at a plane interface, the wave theory of Huygens will always give a light path that takes the shortest time to travel between any two points.

31. Measuring the Speed of Light

Suppose we observe some periodic process occurring at some distance from us. For definiteness we will consider a moon going around a distant planet, but the analysis below would apply to any process that repeats periodically. Suppose that the moon reaches the same stage in its orbit at two consecutive times
t
₁
and
t
₂
; for instance these might be times that the moon consecutively emerges from behind the planet. If the intrinsic orbital period of the moon is
T
, then
t
₂
–
t
₁
=
T.
This is the period we observe, provided the distance between us and the planet is fixed. But if this distance is changing, then the period we observe will be shifted from
T
, by an amount that depends on the speed of light.