Question

The pupil of the eye is 3mm in diameter. Find the angular resolution of the eye for the light of wavelength $5461{A^ \circ }$.
A. 0.7640 min
B. 0.5852 min
C. 0.6257 min
D. 0.4589 min

Answer 1

Hint: Using the formula of angular resolution $\Delta \theta = \dfrac{{0.61.\lambda }}{a}rad$ where $\lambda $ is the wavelength of the light and $2a$ is the diameter of the circular aperture or the diameter of the lens, whichever is smaller, we will get the required solution.

Complete step by step answer:

We know that the diameter of the pupil of the eye is always less than the diameter of the eye lens. Therefore,
$ \Rightarrow 2a = 3mm = 3 \times {10^{ - 3}}m$
Calculating the value of$a$ from the above by dividing both sides by 2. We get,
$ \Rightarrow a = 1.5 \times {10^{ - 3}}m$
Given the wavelength of the incoming light,
$ \Rightarrow \lambda = 5461{A^ \circ }$
Converting the units to m, we get,
$ \Rightarrow \lambda = 5461 \times {10^{ - 10}}m$
Using the formula for calculation of angular resolution,
$ \Rightarrow \Delta \theta = \dfrac{{0.61\lambda }}{a}$
We put the values of wavelength and radius of aperture in m as calculated before into the above equation,
$ \Rightarrow \Delta \theta = \dfrac{{0.61 \times 5461 \times {{10}^{ - 10}}}}{{1.5 \times {{10}^{ - 3}}}}$
Solving the multiplication in the numerator and combining the power of 10. We get,
$ \Rightarrow \Delta \theta = \dfrac{{3331.21 \times {{10}^{ - 10 + 3}}}}{{1.5}}$
Calculating the above we get,
$ \Rightarrow \Delta \theta = 2221 \times {10^{ - 7}} = 2.221 \times {10^{ - 4}}rad$
We know that, $1rad = \dfrac{{180}}{\pi }\deg $ , hence, we get
$ \Rightarrow \Delta \theta = 2.221 \times {10^{ - 4}} \times \dfrac{{180}}{\pi }\deg $
Approximating, $\pi \approx 3.14$ , we get
$ \Rightarrow \Delta \theta = \dfrac{{2.221 \times 180}}{{3.14}} \times {10^{ - 4}} = 127.32 \times {10^{ - 4}}\deg $
We know that, $1\deg = 60\min $.
Hence, we get,
$ \Rightarrow \Delta \theta = 127.32 \times 60 \times {10^{ - 4}} = 7639.2 \times {10^{ - 4}}\min $
Hence, we get the exact value for angular resolution as,
$ \Rightarrow \Delta \theta = 0.7639\min \approx 0.7640\min $
Therefore the correct option is A.

Note: Although it is a formula based question, we need to remember that the unit of angular resolution is Radians and not Degrees and we need to convert the unit to minutes appropriately. This is a scope for error as there is no dimension for angles and the formula is just the ratio of lengths so it doesn’t provide the hint. Also, all the options units are in minutes of arc which do not provide the necessary hint in this regard as well.