lecture17

lecture17 - 17. Lecture 17 17.1 Concave mirror Continuing...

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Unformatted text preview: 17. Lecture 17 17.1 Concave mirror Continuing with our study of the concave mirror we can find now where the image is formed. In the approximation of small angles we are using, all the rays of light emanating from the object that are reflected in the mirror intersect at a point which is the position of the image. To find such position we need to us at least two rays. This is illustrated in fig.98. One is the ray that is parallel to the horizontal axis that is reflected toward the focal point. Another one is the one that goes through the focal point which is reflected parallel to the horizontal axis. This is so because the path of the light can always be backtracked, so if it arrives parallel then goes to the focus implies that if it comes from the focus it reflects parallel. Where these two rays intersect is where the image is located. Another ray that can be used is the one that hits the center of the mirror which reflect with the same angle as it arrives. This is because at that point the normal to the mirror is horizontal so a ray hitting the center behaves as if it were reflecting from a vertical flat mirror. Just to be clear when we say “horizontal” or “vertical” we are assuming that the axis of the mirror is the horizontal direction in the picture. From the figure we see that, in this case, the image is real, namely it is formed in front of the mirror. On the other hand, looking at figure 99 we find that, if the object is closer to the mirror than the focal distance f , then the image is virtual. In the next section we will be more precise and find a mathematical equation that determines the position and height of the image. Figure 98: Image of an object by a concave mirror when the object is further than the focal point. Follow the rays and see how the image is constructed. The highlighted triangles are similar to each other and used in deriving the mirror equation. – 112 – Figure 99: Image of an object by a concave mirror when the object is closer than the focal point. 17.2 Mirror equation Consider the case of the concave mirror that we have already discussed and illustrated in fig.98. If we put an object of height h at a distance s from the mirror we are interested in finding the height h￿ and position s￿ of the image. Before starting let us discuss how we set up the sign conventions. We are going to take s￿ > 0 if the image is formed on the same side as the object (real image) and s￿ < 0 if it is formed behind the mirror (virtual image). Furthermore, if the image is upright we take h￿ > 0 and if it is inverted we take h￿ < 0. We can now proceed to find the position and height of the image. From fig.98, using that the highlighted triangles are similar (namely they have the same angles and their sides are proportional) we find that: h −h￿ = s−f f (17.1) Here we took into account that h￿ < 0 so the length of the corresponding side of the triangle is actually −h￿ . If we do the same with the image we find another equality: − h￿ h = ￿−f s f – 113 – (17.2) which actually is the same as the previous equation after interchanging s ↔ s￿ , h ↔ −h￿ . From eq.(17.1) we find hf h￿ = − (17.3) s−f replacing this value of h￿ in the eq.(17.2) we find: hf h = (s￿ − f )(s − f ) f ⇒ hf h ( s￿ − f ) = s−f f ⇒ s￿ = f2 +f s−f (17.4) Taking common denominator in the expression for s￿ we finally find sf s−f hf h￿ = − s−f s￿ = (17.5) (17.6) where we included eq.(17.3). These two equations completely determine the position and size of the image. Although we derived them for the case where s > f , it is straightforward to check they also work when s < f . It is conventional to rewrite them in a way that makes more evident the symmetry between the object and the image: s￿ = sf s−f ⇒ 1 s−f 11 = =− ￿ s sf f s ⇒ 1 1 1 + ￿= ss f (17.7) For the height we have: h￿ hf s − f h =− =− ￿ s s − f sf s So we get the equivalent equations: 1 1 1 + ￿= ss f ￿ −h h = ￿ s s (17.8) (17.9) (17.10) 17.3 Convex mirror A convex mirror is shaped as the exterior of a sphere. Rays which arrive parallel to the axis are divergent after being reflected. They appear to originate from a point behind the mirror called the focal point. On the other hand, a ray that is directed toward the focal point will be reflected parallel. In fig.100 we see that in this case the image is always virtual and upright. With some algebra and using the same ideas as before we can derive that the convex mirror obeys the same mirror equation as the concave one with the only change being that now f < 0. – 114 – Figure 100: Image of an object by a convex mirror. The image is always virtual and upright. In this case f < 0. 17.4 Convergent lens Another important optical component is the lens. In the ideal situation, rays arriving parallel to the axis of the lens are transmitted and converge into a single point called the focal point. The lens works using the laws of refraction. Typically it is made of glass or a similar material. The rays of light are refracted twice, once in each surface of the lens and that produces the desired effect. As with the mirror, the behavior of the lens is only approximate and valid for rays which are not too far from the axis and/or arrive at small angles. If the rays arrive from the other side of the lens they will converge into another focal point situated at the same distance as the one we already discussed but on the other side of the lens. Furthermore, if a ray goes through a focal point, it will become parallel to the axis after crossing the lens. Finally, a ray going through the center of the lens is transmitted right through. The rules to construct images are similar as with the mirrors. In the figures 101 and 102 we see two cases, one where the object is far from the lens and the other when it is closer that f . – 115 – Figure 101: Image of an object by a convergent lens when the object is further than the focal point. Figure 102: Image of an object by a convergent lens when the object is closer than the focal point. – 116 – Figure 103: Demo illustrating the properties of lenses – 117 – ...
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