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Fermat Stationary Time

In 1658, Fermat enunciated the principle of least time for the path followed by light radiations. According to this principle, the path actually taken by a light is passing from one point to the other is the path of least time. The fundamental laws of rectilinear propagation, reflection and refraction can be derived from this principle. However, in some cases it has been shown that the time taken by light is not minimum but maximum. Therefore, in the modified form, Fermat’s principle of least time is known as Fermat’s principle of stationary time. According to this principle, the path taken by a ray of light in passing from one point to the other is the path of minimum or maximum time.

(i) Rectilinear propagation of light: As a straight line is the path of least distance between two points, therefore, the time taken by the ray of light along a straight line is minimum as compared to some other path which may be curved. Therefore, Fermat’s principle shows that light rays travel along straight line paths in a homogeneous medium.

(ii) Derivation of the laws of reflection: Consider a plane mirror and a plane ABCD normal to it. The incident ray AO is reflected, along OB.ON is the normal. The rays, AO, OB and ON lie in the plane ABCD. Suppose a light ray is reflected from a point P where OP is normal to the plane ABCD. It means the incident ray AP is reflected along PB. Here, AP > AO and PB > OB. Therefore, the path AP + PB is greater than AO + OB, which is against Fermat’s principle. Therefore, P must coincide with O for the path to be minimum. Hence, AO, OB and ON all lie in the same plane.

Suppose, the angle of incidence is i and the angle of reflection is r. The point O is to be located along CD.

Take AO = u, OB = v, AD = m, BC = n and velocity of light = c.

Time taken along AOB = t = (u+v)/c

In the Δ AOD, sec i = OA/AD = u/m or u = m sec i

In the Δ BOC, sec r = OB/BC = v/n or v = n sec r

∴ t = 1/c (m sec i + n sec r)                     (i)

If the point O is shifted, there is a slight change in i and r. Differentiating equation (i)

dt = 1/c (m sec i tan i di + n sec r tan r dr)

For the time to be minimum, dt = 0

∴ 1/c = (m sec i tan i di + n sec r tan r dr) = 0

or, m sec i tan i di = –n sec r tan r dr                    (ii)

Whatever may be the position of O, DO + OC = constant

But DO = m tan i and OC = n tan r

∴ m tan i + n tan r = constant

Differentiating,

m sec² i di + n sec² r dr = 0

or, m sec² i di = –n sec² r dr                             (iii)

Dividing (ii) by (iii),

sin i = sin r or i = r

Therefore, (i) the angle of incidence is equal to the angle of reflection and (ii) the incidence ray, reflected ray and the normal lie in the same plane.