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lecture18 - 18. Lecture 18 18.1 Lens equation Going back to...

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Unformatted text preview: 18. Lecture 18 18.1 Lens equation Going back to the convergent lens, we can once again find an equation that determines the position and size of the image given the position and size of the object. The sign convention for the image’s position is now that s￿ > 0 if the image is on opposite sides of the lens as the object and s￿ < 0 if the image is on the same side of the object. This makes sense because we look at the lens from behind whereas we looked at the mirror from the front. That is a real image has s￿ > 0 in both cases. With these conventions and looking at figure 101 we can derive, similarly as for the mirror: h −h￿ = s−f f (18.1) − h￿ h = ￿−f s f (18.2) and also Since the equations are the same as for the mirror we can immediately derive sf s−f hf h￿ = − s−f s￿ = (18.3) (18.4) or equivalently: 1 1 1 + ￿= ss f ￿ −h h = ￿ s s (18.5) (18.6) It is easy to see that once again these equations are valid if s > f or s < f . 18.2 Divergent Lens When rays arrive parallel to the axis into a divergent lens they become divergent and seem to originate from a focal point situated on the same side of the lens from where the rays arrived. As seen in fig. 104 the image is always virtual and up-right. The equation for the lens can be applied to this case if we take f < 0. – 118 – Figure 104: Image of an object by a divergent lens. It is always virtual and upright. 18.3 Camera, microscope, telescope There are numerous application of lenses and mirrors. The photographic camera is one of them. It actually works on the same principle as our eye. It creates an image on a surface sensitive to light. The image is then read electronically or chemically by producing a reaction on a film. The minimal setup is a convergent lens which creates a real image as in fig.101. The screen is positioned at the point where the image is formed. Another application is the microscope. The minimal setup in this case is a convergent lens as in fig.102. Actually this is also the usual magnifying glass. Actual microscopes have more lenses that increase the magnification but work on the same principle. Finally we have the telescope which is used to look at objects far away. When a point is far away, the rays coming from such point, for all practical purposes are parallel to each other. On the other hand if we have an extended object far away, and the object is large enough, the rays originating from different points of the object arrive with different angles. In that way we can talk about the angular size of an object which is the difference in angles between rays arriving from opposite edges of the object. As seen in figure 105, out of parallel rays, a convergent lens forms an image on its focal plane. This is how our eyes form images for objects at infinity. The angular size of the object is determines the size of the image on our retina. In that way we see for – 119 – example the Moon as an object which has an apparent size which does not seem too big compared to everyday objects even if we know it is much larger than any object we can handle. The reason of course being that the Moon is much further away that any everyday object. If, as in figure 105 we introduce another lens whose focal point coincides with the one of the first lens, the rays will once again be parallel but now the angular size will be different. We want the angular size to be larger so the object will appear larger and we will be able to see more details. From fig.105 we can derive that tan θ = h , f tan θ￿ = h f￿ (18.7) where h is the height of the image created by the first lens and f , f ￿ are the focal distances of both lenses. The angles θ and θ￿ are the angles at which a given ray arrives and then emerges from the telescope. Since we always consider small angles we have tan θ ￿ θ, tan θ￿ ￿ θ￿ (18.8) which implies θ￿ f =￿ (18.9) θ f which is defined as the magnification of the telescope. If we have f ￿ < f then we also have θ￿ > θ as we wanted. magnification = 18.4 Aberrations We insist once again that all the equations and image formation we nave studied for mirrors and lenses are approximations for small angles. Sometimes these are called the laws of ideal mirrors and lenses whereas actual ones do not behave exactly the same. The differences between the ideal and actual behavior are called aberrations. Let us discuss two simple ones. First, we said that a spherical mirror forms a sharp image only if the rays arrive close to the axis. namely the mirror is small compared to the radius of curvature. This is illustrated in fig.106 where we see that the rays emanating from the object do not all cross at the same point. For rays reaching the mirror further from the axis this is evident. This is called spherical aberration. If we put a screen or photographic plate at the position of the image the image will be blurred for such a big mirror. On the other hand, if we just take a look from the left with our own eyes this might not be so evident because our eyes are quite small and will only capture a few of the rays. Such rays would intersect at the image point and the image will appear sharp essentially because our eyes are taking advantage only of the central part of the mirror. – 120 – Figure 105: Simple refractive telescope. Two lenses increase the angular size of objects situated at infinity Another kind of aberration is called color or chromatic aberration. It applies only to lenses. Although we did not discuss this in detail it turns out that the index of refraction is different for different colors as seen in a prism which separate the different colors from white light. A lens works similarly and therefore the focal point is slightly different for different colors. This distorts the images produced by lenses and needs to be corrected in some applications by putting several lenses such that their color aberrations cancel each other. – 121 – Figure 106: Spherical mirror. Images formed by a big mirror (compared with its curvature radius) are not sharp. – 122 – ...
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