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: