Images from Spherical Mirrors — Question 9

Question 9

Moving a curved mirror shifts its image by more than the mirror displacement

Use the paraxial mirror relations $1/p+1/q=1/f$ and $m=-q/p$, with p positive for a real object, q positive for a real image, and f positive for a concave mirror. A fixed object is at laboratory coordinate x=0. A concave mirror at x=X faces the object, with constant focal length f and X greater than f. Its real image has laboratory coordinate $x_I=X-q$, where p=X.

1. Derive the image velocity for a slowly translating mirror.

2. Evaluate the velocity ratio when X=3f, and compare with the plane-mirror limit at fixed X.

Question 9 - Solution

Moving a curved mirror shifts its image by more than the mirror displacement

(a) Method and calculation

$$q=\frac{fX}{X-f},\quad\frac{dq}{dX}=-\frac{f^2}{(X-f)^2}.$$$$\boxed{v_I=\dot X\left[1+\frac{f^2}{(X-f)^2}\right]}.$$

(b) Interpretation and further calculation

$$X=3f\Rightarrow\boxed{v_I/\dot X=5/4}.$$$$|f|\to\infty\ \text{at fixed }X:\quad\boxed{v_I/\dot X\to2}.$$

Physical check

The plane limit continues the signed mirror formula to a virtual image; image coordinates must be converted back to the laboratory frame.