Reflection
We shall first
examine a way of deriving the law of reflection
In 1650 Pierre
Fermat discovered a remarkable principle, which we can express in
these terms:
A light ray
travelling form one fixed point to another fixed point follows a
path such that, compared with nearby paths, the time required is
either a minimum or a maximum or remains unchanged that is stationary).
We can readily
derive the law of reflection from this principle. Figure 10 shows
two fixed points A and B and a reflecting ray APB connecting them
(We assume that ray APB lies in the plane of the figure). The total
length L of this ray is
L= square root
of (a2 +x2)+ square root of (b2
+(d-x)2)
where x locates
the point P at which they ray touches the mirror.
According to
Fermat's principle, P will have a position such that the time of
travel t=L/c of the light must be a minimum (or a maximum or must
remain unchanged), which occurs when dt/dx=0. Taking this derivative
yields
Dt/dx=1/c dL/dx
=1/2c(a2+x2)-1/2
(2x)+1/2c [b2+ (d-x)2]-1/2 (2)(d-x)(x-1)=0
which we can
rewrite as
x/(square root
of (a2+ x2))=(d-x)/ (square root of b2+
(d-x)2)
(In evaluating
the derivative, note that we hold the endpoints fixed and vary the
endpoints fixed and vary the path by allowing x to vary.)
Comparison with
Fig 10 shows that we can rewrite this as
sinq1=sinq'1,
Or
q1=q'1
which is the
law of reflection
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Total Internal Reflection Applet
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The Optics Book