In Fig. 27-2, for example, s, s, and R are all positive; in Fig. 27-3, s and R are positive, but sis negative. If we had used a concave surface, our formula(27.3 would still give the correct result if we merely make r a negative quantity n working out the corresponding formula for a mirror, using the above conventions, you will find that if you put n =-I throughout the formula(27. 3) as though the material behind the mirror had an index-1), the right formula for a mirror results! Although the derivation of formula(27. 3)is simple and elegant, using least time, one can of course work out the same formula using Snell's law, remembering that the angles are so small that the sines of angles can be replaced by the angles themselves 27-3 The focal length of a lens Now we go on to consider another situation, a very practical one. Most of the lenses that we use have two surfaces, not just one. How does this affec matters? Suppose that we have two surfaces of different curvature, with glass filling the space between them(Fig. 27-5). We want to study the problem of focusing from a point O to an alternate point O'. How can we do that? The answer is this: First, use formula(27.3)for the first surface, forgetting about the second surface. This will tell us that the light which was diverging from O will appear to be converging or diverging, depending on the sign, from some other point, say O. Now we consider a new problem. We have a different surface. between glass and air, in which rays are converging toward a certain point o'. Where will they actually converge? We use the same formula again! We find that they con Fig. 27-5. Image formation by a verge at o. Thus, if necessary, we can go through 75 surfaces by just using the wo-surface lens same formula in succession, from one to the next There are some rather high-class formulas that would save us considerable energy in the few times in our lives that we might have to chase the light through five surfaces, but it is easier just to chase it through five surfaces when the problem arises than it is to memorize a lot of formulas, because it may be we will never have to chase it through any surfaces at all! any case, the principle is that when we go through one surface we find a new position, a new focal point, and then take that point as the starting point for the next surface, and so on. In order to actually do this, since on the second surface we are going from n to I rather than from 1 to n, and since in many systems there is more than one kind of glass, so that there are indices n1, n2, e really need o' a generalization of formula(27. 3) for a case where there are two different indices, nI and n2, rather than only n. Then it is not dificult to prove that the general form of (27. 3)is R Fig. 27-6. A thin lens with two posi- Particularly simple is the special case in which the two surfaces are very close tive rad together-so close that we may ignore small errors due to the thickness. If we draw the lens as shown in Fig. 27-6, we may ask this question: How must the lens be built so as to focus light from o to o? Suppose the light comes exactly to the edge of the lens, at point P. Then the excess time in going from o to Ois (n h 7 25)+(n h2/2s"), ignoring for a moment the presence of the thickness t of glass of index n2. Now, to make the time for the direct path equal to that fo the path OPO, we have to use a piece of glass whose thickness Tat the center is such that the delay introduced in going through this thickness is enough to compensate for the excess time above. Therefore the thickness of the lens at the center must be given by the relationship (n1h2/2)+(n1h2/2)=(m2-n1)7 (278) We can also express T in terms of the radii R, and r2 of the two surfaces. Paying attention to our convention (3), we thus find, for R1< R2(a convex lens), T=(h2/2R1)-(h2/2R2)