Two Types of Image — Question 9

Question 9

An eye can receive a virtual image with a definite angular size

A virtual upright image of height H lies distance d behind a transparent output plane. An eye lies distance e in front of that plane and receives the diverging rays. Both distances are positive and H is much smaller than d+e.

1. Find the small-angle angular height seen by the eye.

2. If H=12 mm, d=240 mm, and e=60 mm, find the angular height and its ratio to the value at e=0.

Question 9 - Solution

An eye can receive a virtual image with a definite angular size

(a) Method and calculation

$$\ell_{\mathrm{eye,image}}=d+e,\quad\alpha=\tan^{-1}[H/(d+e)].$$$$\boxed{\alpha\simeq H/(d+e)}.$$

(b) Interpretation and further calculation

$$\boxed{\alpha\simeq12/300=0.0400\,\mathrm{rad}}.$$$$\boxed{\alpha(e)/\alpha(0)\simeq d/(d+e)=0.80}.$$

Physical check

The eye traces the received rays backward to the virtual image; the relevant viewing distance includes its own separation from the output plane.