goes out, point O'comes in, and vice versa. As point O goes toward infinity, point O' keeps moving in until it reaches a certain distance, called the focal length inside the material. If parallel rays come in, they will meet the axis at a distance f. Likewise, we could imagine it the other way.( Remember the reciprocity rule if light will go from O to O, of course it will also go from O to O. )Therefore, if we had a light source inside the glass, we might want to know where the focus is In particular, if the light in the glass were at infinity(same problem)where wor it come to a focus outside? This distance is called f. Of course, we can also put it the other way. If we had a light source atf and the light went through the surface, then it would go out as a parallel beam. We can easily find out what fandf'are ∥=(n-1)/R f=Rn/(n-1), L/f=(n-1)/R f=R/(n-1) We see an interesting thing: if we divide each focal length by the corresponding dex of refraction we get the same result! This theorem, in fact, is general. It is true of any system of lenses, no matter how complicated, so it is interesting remember. We did not prove here that it is general-we merely noted it for a single are related in this way. Sometimes Eq.(27. 3)is written in the form t a syster surface, but it happens to be true in general that the two focal lengths of a system This is more useful than (27. 3) because we can measure more easily than we can measure the curvature and index of refraction of the lens if we are not interested in designing a lens or in knowing how it got that way, but simply lift it off a shelf, the interesting quantity is f, not the n and the l and the r Now an interesting situation occurs if s becomes less than f. what happens en? If s <f then (1/s)>(l/, and therefore s' is negative; our equation says that the light will focus only with a negative value of s, whatever that means It does mean something very interesting and very definite. It is still a useful formula 等转。k in other words, even when the numbers are negative. what it means is shown in o' Fig. 27-3. If we draw the rays which are diverging from O, they will be bent, it is rue, at the surface, and they will not come to a focus, because o is so close in that they are"beyond parallel. However, they diverge as if they had come from a point o outside the glass. This is an apparent image, sometimes called a virtual Fig. 27-3. A virtual image image. The image O' in Fig. 27-2 is called a real image. If the light really comes to a point, it is a real image. But if the light appears to be coming from a point, a fictitious point different from the original point, it is a virtual image. So when s' comes out negative, it means that o' is on the other side of the surface, and every- ling is all right Now consider the interesting case where R is equal to infinity; then we have (1/s)+(n/s)=0. In other words, s'=-ns, which means that if we look from a dense medium into a rare medium and see a point in the rare medium, it appears to be deeper by a factor n. Likewise, Ise the same equation backwards, so that if we look into a plane surface at an object that is at a certain distance inside the dense medium, it will appear as though the light is coming from not as far back(Fig. 27-4). When we look at the bottom of a swimming pool from above it does not look as deep as it really is, by a factor 3/4, which is the reciprocal of the Fig. 27-4. A plane surface re-images index of refraction of water the light from o to o We could go on, of course, to discuss the spherical mirror. But if one appreci ates the ideas involved, he should be able to work it out for himself. Therefore we leave it to the student to work out the formula for the spherical mirror, but we mention that it is well to adopt certain conventions concerning the distances involved (1)The object distance s is positive if the point O is to the left of the surface (2)The image distance s' is positive if the point o' is to the right of the surface (3)The radius of curvature of the surface is positive if the center is to the right of the surface