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3.9 A final note on Fermat's principle

Fermat was ahead of his time in formulating a law of Nature on the basis of a minimum principle. Some years later Maupertius (1698-1759) enunciated a more general law, Nature is thrifty in all its actions', but this was a somewhat qualitative statement with much less supportive experimental evidence compared with Fermat's principle of least time for light.

The pioneers of generalised classical mechanics

A quantitative mathematical theory was subsequently developed over a period of a century by Leonhard Euler (1707-1783), Joseph Louis Lagrange (1736-1813) and William Rowan Hamilton (1805-1865). It was based on a statement known as the principle of least action. This principle applied not only to light but also to all mechanical systems, and stated that every system develops with time in such a way that a certain mathematical quantity called the action is a minimum. This forms the basis of a generalised theory of mechanics, which includes Newton's mechanics, and puts it on a more fundamental footing. We now know that the principle of least action applies also to relativity theory and quantum mechanics.

Appendix 3.1 The lifeguard problem

To find point P

Let us specify the position of P by its distance x from the foot of the perpendicular dropped from A, as shown in Figure 3.25.

The total time T for the journey can be expressed in terms of the fixed parameters vp v2, h1, h2, and d as well as the only parameter over which the lifeguard has some control, namely x:

Figure 3.25 The lifeguard's path.

i : angle of incidence r : angle of reflection

Figure 3.25 The lifeguard's path.

To calculate the minimum time, set the first derivative of time with respect to x equal to zero:

— = Sin 1 (Formula for the quickest route) v 2 sin r

The lifeguard will have to calculate the optimum position of P such that the angles i and r satisfy the above relation!

Note that in the calculation given above we expressed the time as a function of the variable x and then equated the derivative dT/dx to zero. This gave us the stationary point of the function which could be a maximum, minimum or saddle point. To determine which it is, mathematically, we could obtain the second derivative and determine whether that is negative, positive or zero at that point. Physically it is obvious that it is in fact a minimum in this example. A more precise statement of Fermat's principle refers to stationary rather than minimum values of the time.

x h2

Appendix 3.2 The lens equation

We can use Figure 3.26 to relate object and image distances to show how the size of the image is related to that of the object. From similar triangles COO' and CII':

From similar triangles F'BC and F'II':

Combining equations (3.1) and (3.2)

Dividing by uvf :

We can obtain the same result from similar triangles FOO' and DCF.

Newton's equation

If we measure distances from the focal point instead of from the lens, we get a very neat relationship known as the Newtonian form of the thin lens equation.

Using Figure 3.26:

Object distance from focus, x = O'F Image distance from focus, x' = I'F'

h h' h' x x f h f h_ h h_ x' _ f^ h _ f xx' _ f (Lens equation in Newtonian form)

Appendix 3.3 Calculating the power of spectacles

(We can assume that the spectacle lens and the eye lens are in contact.)

A healthy young adult can accommodate the eye so that it focuses on objects 25 cm away (the near point). The power P of the fully accommodated eye can be calculated from the lens equation:

P _ — _ — +1 _ + _ 4 + 50 _ 54 diopters f u v 0.25 0.02

A patient with a sight defect can see objects clearly only when they are at a distance of between 0.8 m and 2.5 m from his eyes. What is the power of the lens which will extend his clear vision to the normal range of 0.25 m to infinity?

The problem here is presbyopia, described in Section 3.6. We need to prescribe separate glasses for reading and for distant vision.

Reading. The patient needs spectacles to aid accommodation and restore his near point to 0.25 m, as illustrated in Figure 3.27.

Normal eye: Fully accommodated power = 54 diopters (Example 1).

Figure 3.27 Reading glasses (not drawn to scale).

Figure 3.27 Reading glasses (not drawn to scale).

Figure 3.28 'Distance' glasses (not drawn to scale).

Figure 3.28 'Distance' glasses (not drawn to scale).

Patient's eye:

The spectacles must be fitted with convex (converging) lenses with a power of 54-51.25 = 2.75 diopters to restore his near point to 0.25 m.

Distant vision. The normal eye focuses an object at infinity on the retina when fully relaxed. The defective eye focuses an object at infinity in front of the retina (it is still 'too strong') when fully relaxed.

Normal eye: 'Fully relaxed' power = 50 diopters.

Defective eye:

'Fully relaxed' power = — +1 = -1: + t^t = 50.4 diopters.

The spectacles must be fitted with concave (diverging) lenses to restore his far point to infinity, as shown in Figure 3.28. The power of the lenses is 50 - 50.4 = -0.4 diopters.

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