Question

Identical objects are located at the same distance from two spherical mirrors, A and B. The magnifications produced by the mirrors are mA = 6.0 and mB = 2.7. Find the ratio fA/fB of the focal lengths of the mirrors.

Answer 1

By mirror formula, 1/v + 1/u = 1/f

where u = object distance

v = image distance

f = focal length of mirror

also, magnification(m) = -v/u

for object A,

mA = -vA/uA

vA = -6*uA

So, 1/fA = 1/vA + 1/uA

1/fA = -1/(6*uA) + 1/uA           eq(1)

for object B,

mB = -vB/uB

vB = -2.7*uB

So, 1/fB = 1/vB + 1/uB

1/fB = -1/(2.7*uB) + 1/uB            eq(2)

Now divide eq (2)/eq(1),

fA/fB = [-1/(2.7*uB) + 1/uB]/[-1/(6*uA) + 1/uA]

Also given that both objects are located at same distance, So

uA = uB

Using above relation

fA/fB = [-1/2.7 + 1]/[-1/6 + 1]

fA/fB = (1.7/2.7)/(5/6) = 1.7*6/(2.7*5)

fA/fB = 0.76

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