Question

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The same object is located at the same distance from two spherical mirrors, A and B. The magnifications produced by the mirrors are mA = 3.9 and mB = 2.5. Find the ratio fA/fB of the focal lengths of the mirrors.

Answer 1

The equation for the magnification m of an object is , where is the image distance and s is the object distance. The mirror equation is given by , where f is the focal length of the mirror. The goal is to use these two equations to find the focal lengths of each mirror and then take the ratio.

Step 1) First use the magnification equation for the first mirror (mirror on the left, A). Since this means , or . Assume that s is positive by recalling when the object is on the same side as the incoming light the object distance is positive. Since the magnification is positive too, it must mean that the image distance is negative.

Step 2) Next plug into the mirror equation and simplify.

Step 3) Next use the magnification equation for the second mirror (mirror on the right, B). Since this means or . Assume that s is positive by recalling when the object is on the same side as the incoming light the object distance is positive. Since the magnification is positive too, it must mean that the image distance is negative.

Step 4) Next plug into the mirror equation and simplify.

Step 5) Finally take the ratio of the focal lengths by dividing by .

The ratio of the focal lengths is 0.81.