Question

A spherical mirror is polished on both sides. When the convex side is used as a mirror, the magnification is 0.25. What is the magnification when the concave side is used as a mirror, the object remaining the same distance from the mirror?

Answer 1

The magnification equation states m = −d_i/d_o, so 

d_i = −(0.25)(d_o)

The lens equation states 1/d_o + 1/d_i = 1/f, so

1/d_o + 1/−(0.25)(d_o) = 1/f

⇒ 1/d_o + −4/d_o = 1/f

⇒ −3/d_o = 1/f

⇒ f = −d_o/3

Now if we flip the mirror the new focal length is simply the original with the sign flipped. Thus, 

f = d_o/3

We can plug this into another lens equation to get 

1/d_o + 1/d_i = 3/d_o

⇒ 1/d_i = 2/d_o

⇒ d_i = d_o/2

Therefore the magnification must be

m = − d_i /d_o = − d_o/2/d_o = −0.5000