Refraction (Continuation)

The law of refraction is n₁sinq₁= n₂sinq₂

To prove it, consider Fig 17 which shows two fixed points A and B in two different media and a refracting ray APB connecting them. The time t for the ray to travel from A to B is given by

t=(L₁/ V₁)+(L₂/ V₂)

Using the relation n=c/v we can write this as

t=[(n₁L₁+ n₂L₂]/c=L/c

where L is the optical path length defined as

L=n₁L₁+ n₂L₂

For any light ray travelling through successive media, the optical path length is the sum of the products of the geometrical path length and the index of refraction of that medium. The equation l_n=l/n shows that the optical wave length is equal to the length that this same number of waves would have if the medium were a vacuum. Do not confuse the optical path length with the geometrical path length which is L₁+ L₂ for the ray of Fig 17.

Fermat's principle requires that the time t for the light to travel the path APB must be a minimum( or a maximum or must remain unchanged) which in turn requires that x be chosen so that dt/dx=0. The optical path length in Fig 17 is

L=n₁L₁+ n₂L₂₌n₁(square root of a²+ x²)+n₂(square root of b²+(d-x)²)

Substituting this result into t=[(n₁L₁+ n₂L₂]/c=L/c and differentiating, we obtain

Dt/dx=1/c(dL/dx)

=n₁/2c(a²+ x^2)-1/2(2x)+ n₂/2c[b²+ (d-x)²]^-1/2 (2)(d-x)(x-1)=0

Which we can write as

n₁ [x/( square root of a²+ x²)]=n₂  [(d-x)/ (square root of b²+(d-x)²)]

FIGURE 17 Credits: Halliday David

Comparision with Fig 17 shows that we can write as n₁sinq₁= n₂sinq₂ which is the law of refraction.