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Question: Aperture of the human eye is \(2mm\). Assuming the mean wavelength of the light to be \(5000\)Å, the...

Aperture of the human eye is 2mm2mm. Assuming the mean wavelength of the light to be 50005000Å, the angular resolution limit of the eye is nearly:
A) 2 minuteA)\text{ 2 minute}
B) 1 minuteB)\text{ 1 minute}
C) 0.5 minuteC)\text{ 0}\text{.5 minute}
D) 1.5 minuteD)\text{ 1}\text{.5 minute}

Explanation

Solution

This problem can be solved by using the direct formula for the angular resolution of a lens in terms of the aperture of the lens and the wavelength of the light used. The lens in this case is the eye lens. Then this value can be converted from radian into the required units.

Formula used:
R=1.22λaR=\dfrac{1.22\lambda }{a}

Complete answer:
We will use the direct formula for the angular resolution limit of a lens to solve this problem. So, let us write the formula.
The angular resolution limit RR of a lens of aperture aa is given by
R=1.22λaR=\dfrac{1.22\lambda }{a} --(1)
Where λ\lambda is the wavelength of the light used.
Now, let us analyze the question.
The aperture of the human eye is given to be a=2mm=2×10−3ma=2mm=2\times {{10}^{-3}}m (∵1mm=10−3m)\left( \because 1mm={{10}^{-3}}m \right)
The wavelength of the light used is λ=5000\lambda =5000Å=5000×10−10m=5×10−7m=5000\times {{10}^{-10}}m=5\times {{10}^{-7}}m
Let the required angular resolution limit be RR.
Therefore, using (1), we get
R=1.22×5×10−72×10−3=3.05×10−4radR=\dfrac{1.22\times 5\times {{10}^{-7}}}{2\times {{10}^{-3}}}=3.05\times {{10}^{-4}}rad
∴R=3.05×10−4rad=3.05×10−4×1800π=0.01750\therefore R=3.05\times {{10}^{-4}}rad=3.05\times {{10}^{-4}}\times \dfrac{{{180}^{0}}}{\pi }={{0.0175}^{0}} (∵1rad=1800π)\left( \because 1rad=\dfrac{{{180}^{0}}}{\pi } \right)
∴R=0.01750×60 minute = 1.05 minute ≈1 minute\therefore R={{0.0175}^{0}}\times 60\text{ minute = }1.05\text{ minute }\approx 1\text{ minute} (10=60 minute)\left( {{1}^{0}}=60\text{ minute} \right)
Therefore, the required value of the angular resolution limit is 1 minute1\text{ minute}.

So, the correct answer is “Option B”.

Note:
Students must have gotten an understanding of the concept that for a lens with a greater aperture, the angular resolution limit is more. This is the reason why astronomical telescopes and observatories have a lens of very large diameter or large aperture. This ensures that the angular resolution limit is very high and the telescope can clearly resolve two stars that seem to be very close together due to the small difference in the angles subtended by them on the telescope.