Ch0229 - Chapter 29 Thin Lenses: Lens Equation, Optical...

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Chapter 29 Thin Lenses: Lens Equation, Optical Power 257 29 Thin Lenses: Lens Equation, Optical Power From the thin lens ray-tracing methods developed in the last chapter, we can derive algebraic expressions relating quantities such as object distance, focal length, image distance, and magnification. Consider for instance the case of a converging lens with an object more distant from the plane of the lens than the focal point is. Here’s the diagram from the last chapter. In this copy, I have shaded two triangles in order to call your attention to them. Also, I have labeled the sides of those two triangles with their lengths. By inspection, the two shaded triangles are similar to each other. As such, the ratios of corresponding sides are equal. Thus: o i h h = | | Recall the conventions stated in the last chapter: Physical Quantity Symbol Sign Convention focal length f + for converging lens for diverging lens image distance i + for real image for virtual image image height h + for erect image for inverted image magnification M + for erect image for inverted image In the case at hand, we have an inverted image, so h ′ is negative, so | h ′| = h ′. Thus, the equation i h h = | | can be written as i h h = , or, as F F I II II I III III | h | h o i i
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Chapter 29 Thin Lenses: Lens Equation, Optical Power 258 o i h h = But h h is, by definition, the magnification. Thus, we can write the magnification as: i M = (29-1) Here’s another copy of the same diagram with another triangle shaded. By inspection, that shaded triangle is similar to the triangle that is shaded in the following copy of the same diagram: Using the fact that the ratios of corresponding sides of similar triangles are equal, we set the ratio of the two top sides (one from each triangle) equal to the ratio of the two right sides: | | | | h h h + = f F F I II II I III III | h | h o i F F I II II I III III | h | h i h + | h | | h |
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Chapter 29 Thin Lenses: Lens Equation, Optical Power 259 Again, since the image is upside down, h ′ is negative so | h ′| = h ′. Thus, h h h = f o h h = 1 From our first pair of similar triangles we found that i h h = which can be written i h h = Substituting this into the expression h h = 1 which we just found, we have = i 1 Dividing both sides by and simplifying yields: i 1 1 1 + = (29-2) This equation is referred to as the lens equation . Together with our definition of the magnification h h M = , the expression we derived for the magnification i M = , and our conventions: Physical Quantity Symbol Sign Convention focal length f + for converging lens for diverging lens image distance i + for real image for virtual image image height h + for erect image for inverted image magnification M + for erect image for inverted image
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This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.

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Ch0229 - Chapter 29 Thin Lenses: Lens Equation, Optical...

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