The Optics Course - Reflection and refraction Tim
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# 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

 FIGURE 10 Credits:

The Optics Course

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