lecture8 - Lecture 8 Notes 07 12 Concave spherical mirrors...

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Lecture 8 Notes: 07 / 12 Concave spherical mirrors and ray diagrams A spherical mirror is a reflective segment of a sphere with a radius of curvature R . It can be convex (outside surface of a sphere) or concave (inside surface). First we will consider a concave spherical mirror. The mirror has a radius R, and the distance from the mirror to the object is p . We will draw the diagram with p > R , but our results will hold true for any value of p . Draw two rays from the object to the mirror: one ray passes through the mirror's center of curvature, and therefore is perpendicular to the mirror at the incidence point and returns directly back to the object. The other ray goes to the center of the mirror, and is reflected symmetrically about the horizontal axis. The rays intersect, forming an image a distance q from the mirror. The object has height h , the image has height h' . From similar triangles, h / (p - R) = h' / (R - q) . Also from similar triangles, h' / q = h / p . This gives us
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Consider the behavior of this equation if the object is located very far away. Then, all the rays come in parallel to the mirror's axis. The rays are all directed to the point q , given by In terms of the focal length, our equation relating p , q and f is Parallel rays are focused to a distance f = R / 2 from the mirror, known as the focal point . The ray diagram for parallel rays, as from a source at infinity, looks like this: In reality, the rays don't all go exactly to the focal point, since the rays that are far away from the axis reflect from points closer to the left than those that come closer to the mirror's center. When an image is formed under some other circumstances, we similarly find that the rays don't all exactly cross. The image produced by a spherical mirror is thus somewhat blurry. This is known as spherical aberration . If the mirror is parabolic, rather than spherical, this problem is corrected, as a parabolic mirror has the property that all parallel rays striking it are reflected to exactly the same point. For this reason, mirrors used in optical instruments such as telescopes are parabolic. But for rays that come in sufficiently close to the mirror's central axis, the performance of a spherical mirror approaches that of a parabolic mirror. Thus, for the purposes of this class, we will use spherical and parabolic mirrors interchangeably. The quantity M = -h' / h = -q / p is known as the magnification . Note that with our sign convention, both p and q in the example above are positive, and M is negative. This means that the image is inverted.
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Some examples: Draw ray diagrams, locate the images and determine magnifications for objects located a distance 3 f , f and 0.5 f away from a concave spherical mirror. The case for p = 3f is somewhat similar to the one discussed above. Now that we know about the focal point, we will draw different rays: one will reflect through the center of the mirror, as before, but the other will start out parallel to the mirror's axis and be reflected through the focal point, as discussed above: The image forms at a distance from the mirror q
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