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PHYSICS 415/416 SUPPLEMENTAL PROBLEMS
Part 1
1A. (Fermat’s Principle)
In Section 2-2 of the text book, Fermat’s Principle is used to derive Snell’s Law. Use a similar analysis to derive the Law of Reflection. Refer to the figure at the right.
(a) Find an expression for the transit time t from point A to point B in terms of x,h,s, and v (the speed of light in the medium).
(b) Note that varying the distance x will result in different path lengths and different transit times from A to B. Therefore, minimize your expression for t in (a) with respect to x.
(c) Show that the resulting equation in (b) can be simplified to yield i = r.
1B. (Thin Lenses)
Show that the minimum separation distance between an object and image formed by a single positive lens is four times its focal length.
(a) Draw a picture showing object, image, and lens. Label the object distance (s) , image distance (s’) , and separation between the object & image (L).
(b) Using the thin lens equation, express L in terms of only s (or only s’).
(c) Minimize L with respect to s (or s’). Show that this gives Lmin = 4f.
(d) Draw a ray trace for this minimum L condition. Find the value of s and s’. Find the lateral magnification. Is the image upright or inverted?
1C. (Bessel's Method)
A lens is moved along an optical axis between a fixed object and a fixed screen. The object and screen are separated by a distance L that is more than four times the focal length f of the lens. Two positions of the lens are found for which an image is in focus on the screen, magnified in one case and reduced in the other. If the two lens positions differ by distance D, show that the focal length is given by f = (L2 - D2)/4L.
To help you in this derivation, I suggest you do the following:
(a) Draw a picture showing the object, image, and a positive lens in two positions. Both of these positions of the lens result in a focused image. Identify and label the picture with:
L - distance between object & imageD - distance between the two positions of the lens
s1 - object distance for lens position 1s’1 - image distance for lens position 1
s2 - object distance for lens position 2s’2 - image distance for lens position 2
(b) Express L in terms of s1, s’2, and D.
(c) Invoke the Principle of Reversibility and show that the expression in (b) can be solved to yield
Explain how the Reversibility Principle allows you to do this.
(d) Express L in terms of s1 and s’1. Combining this expression with the one in (c), show that
(e) Finally use the thin lens equation to show that
There are other ways to derive this equation but the above method is probably the quickest. Whichever way you do it, I want to see a picture as described in (a). This method of measuring the focal length is called Bessel’s method but is also commonly known as the method of conjugate foci. Note that this method only works for positive lenses, and only if the object and image separation distance is greater than four times the focal length.