Chapter 23, Problem 107QAP

To determine

(a)

Three consecutive angles of dark ring from the central bright spot on the retina of the eye

Answer to Problem 107QAP

1^st ring occurs at ${\theta }_1=3.05\times {10}^{-4}rad={0.0175}^o$

2^nd ring occurs at ${\theta }_2=5.58\times {10}^{-4}rad={0.0319}^o$

3^rd ring occurs as ${\theta }_1=8.10\times {10}^{-4}rad={0.0464}^o$

Explanation of Solution

Given:

Diameter of pupil = $d_{pupil}=2mm=2\times {10}^{-3}m$

Wavelength of light = $\lambda =500nm=5\times {10}^{-7}m$

Formula used:

Angle of resolution is defined as,

  $\begin{array}{l}\theta =\frac{m\lambda }{d}(\text{in radian)}\\ \text{for first order, m = 1}\text{.22}\\ \text{for second order, m = 2}\text{.233}\\ \text{for the third order , m = 3}\text{.238}\end{array}$

Calculation:

From above formula,

  $\begin{array}{l}\theta =1.22\frac{\lambda }{d}\\ \text{for first order,}\\ {\theta }_1=1.22\left(\frac{5\times {10}^{-7}m}{2\times {10}^{-3}}\right)=\underset{\_}{3.05\times {10}^{-4}rad}=3.05\times {10}^{-4}rad\left(\frac{{180}^o}{\pi }\right)=\underset{\_}{{0.175}^o}\\ \text{for second order, m = 2}\text{.233, then, }\\ {\theta }_2=2.233\left(\frac{5\times {10}^{-7}m}{2\times {10}^{-3}}\right)=\underset{\_}{5.58\times {10}^{-4}rad}=5.58\times {10}^{-4}rad\left(\frac{{180}^o}{\pi }\right)=\underset{\_}{{0.0319}^o}\\ \text{for third order, m = 3}\text{.238,}\\ {\theta }_3=3.238\left(\frac{5\times {10}^{-7}m}{2\times {10}^{-3}}\right)=\underset{\_}{8.10\times {10}^{-4}rad}=8.10\times {10}^{-4}rad\left(\frac{{180}^o}{\pi }\right)=\underset{\_}{{0.0464}^o}\end{array}$

Conclusion:

Thus, from above discussion, we find the angle of resolution for three consecutive dark rings as,

1^st ring occurs at ${\theta }_1=3.05\times {10}^{-4}rad={0.0175}^o$

2^nd ring occurs at ${\theta }_2=5.58\times {10}^{-4}rad={0.0319}^o$

3^rd ring occurs as ${\theta }_1=8.10\times {10}^{-4}rad={0.0464}^o$

To determine

(b)

The radius of dark rings from the bright spot

Answer to Problem 107QAP

The radius of consecutive three rings are found as,

For 1^st ring, $x_1=7.6\times {10}^{-3}mm$

For 2^nd ring, $x_2=1.4\times {10}^{-2}mm$

For 3^rd ring, $x_3=2.0\times {10}^{-2}mm$

Explanation of Solution

Given:

Angle of resolution for consecutive three dark rings is as,

1^st ring occurs at ${\theta }_1=3.05\times {10}^{-4}rad={0.0175}^o$

2^nd ring occurs at ${\theta }_2=5.58\times {10}^{-4}rad={0.0319}^o$

3^rd ring occurs as ${\theta }_1=8.10\times {10}^{-4}rad={0.0464}^o$

Distance from retina to eye lens = $D=25mm$

Formula used:

  $\begin{array}{l}x=\theta D\\ Here,\theta \text{ is in radian}\text{.}\end{array}$

Calculation:

  $\begin{array}{l}\text{For first ring, }\\ x_1={\theta }_{1}D=3.05\times {10}^{-4}rad\times 25mm=7.6\times {10}^{-3}mm\\ \text{for second rings,}\\ x_2={\theta }_{2}D=5.58\times {10}^{-4}rad\times 25mm=1.4\times {10}^{-2}mm\\ \text{for third rings,}\\ x_1={\theta }_{3}D=8.10\times {10}^{-4}rad\times 25mm=2.0\times {10}^{-2}mm\end{array}$

Conclusion:

Thus, we have found the distance of dark rings from the central bright spot.

To determine

(c)

The explanation about the diffraction of light in our eye

Answer to Problem 107QAP

We cannot observe dark and bright rings because angle of resolution is very small to distinguish the consecutive dark and bright rings for human.

Explanation of Solution

Introduction:

Angle of resolution is inversely proportional to the diameter of aperture of the opening and directly proportional to the wavelength of light. As the diameter of opening increases, angle of resolution will decrease.

From above discussion, we see that the angle of resolution is very small. So, locations of the dark and bright fringes are very close to each other. Human eye is not capable to distinguish these thin rings. They feel it as the continuous light.

Conclusion:

Thus, we do not observe such diffraction of light because of very closeness of the consecutive rings to each other.