Two Types of Image — Question 3

Question 3

Three rays can reveal that a single sharp image does not exist

At an output plane, three rays have equations $y_1=2-0.10x$, $y_2=-2+0.10x$, and $y_3=1-0.04x$, in centimeters. They represent rays from the same object point and travel in +x.

1. Find where the first two rays intersect and evaluate the third ray there.

2. Find the intersections of the third ray with each of the first two and decide whether the bundle has an exact point image.

Question 3 - Solution

Three rays can reveal that a single sharp image does not exist

(a) Method and calculation

$$y_1=y_2\Rightarrow\boxed{x_{12}=20\,\mathrm{cm},\ y_{12}=0}.$$$$\boxed{y_3(20)=0.20\,\mathrm{cm}\ne0}.$$

(b) Interpretation and further calculation

$$y_1=y_3\Rightarrow x_{13}=1/0.06=16.6667\,\mathrm{cm};\quad y_2=y_3\Rightarrow x_{23}=3/0.14=21.4286\,\mathrm{cm}.$$$$\boxed{x_{12}\ne x_{13}\ne x_{23}}\Rightarrow\text{no exact common point image}.$$

Physical check

Intersecting two selected rays is insufficient when another ray from the same object point fails to pass through that intersection.