Three Proofs — Question 2

Question 2

The vertex ray proves mirror magnification independently of a parallel ray

A small object of height hO stands distance p before a spherical mirror. Its paraxial image has signed height hI at distance q. The principal axis is normal to the mirror at the vertex. Use the ray from object tip that strikes the vertex.

1. Use equal angles at the vertex to derive signed transverse magnification.

2. Eliminate q with the mirror equation to express magnification in terms of p and f, then give the object-distance condition for an upright image when f is positive and p is positive.

Question 2 - Solution

The vertex ray proves mirror magnification independently of a parallel ray

(a) Method and calculation

$$\frac{h_O}{p}=-\frac{h_I}{q}\Rightarrow\boxed{m=h_I/h_O=-q/p}.$$$$q=fp/(p-f)\Rightarrow\boxed{m=-f/(p-f)}.$$

(b) Interpretation and further calculation

$$f>0,\ p>0:\quad m>0\Longleftrightarrow p-f<0.$$$$\boxed{0<p<f\Rightarrow m>1\ \text{and}\ q<0}.$$

Physical check

The sign of the image height is part of the proof: omitting it would erase the distinction between inversion and upright orientation.