Question

When dots are placed on a page from a laser printer, they must be close enough so that you do not see the individual dots of ink. To do this, the separation of the dots must be less than Raleigh's criterion. Take the pupil of the eye to be 3.3 mm and the distance from the paper to the eye of 40 cm; find the maximum separation (in cm) of two dots such that they cannot be resolved. (Assume the average wavelength of visible light is 550 nm.) cm How many dots per inch (dpi) does this correspond to? dpi T

Answer 1

Answer:

Rayleigh criteria,

                        

01.22X/D

Given that, wavelength used $\lambda$ = 550 nm = 550 x 10^-9 m (visible light)

and D = 3.3 mm = 3.3 x 10^-3 m

Thus, $\theta$ $\approx$ 1.22 (550 x 10^-9 m) / (3.3 x 10^-3 m) = 2.03 x 10^-4 rad

This is the smallest angle subtended between two dots and yet be resolved by the eyes.

Now, we have the separation angle and the distance bwteen the dots and the eyes is 40 cm, we can find the distance of separation between the two dots using the tangent.

Tan$\theta$ = opposite side/adjacent side

Taking the separating distance as the opposite side and the distance between the dots and the eyes as the adjacent side.

Since the angle is very small, so Tan$\theta$$\approx \theta$

Thus, $\theta$ $\approx$ opposite side/adjacent side

      opposite side $\approx$ adjacent side x $\theta$

                           $\approx$ (40 x 10^-2 m) x 2.03 x 10^-4 rad

                          $\approx$ 81.2 x 10^-6 m or 81.2 x 10^-4 cm

Hence, this is the maximum speration distance between the two dots .

(b) One inch = 2.54 cm

Therefore, dots per inch (dpi) = 2.54 cm/81.2 x 10^-4 cm = 312.80 $\approx$ 312 dpi