© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Understand the Problem
We want to find the minimum separation between two points that can be resolved by a human eye. The human eye is modeled as an aperture (the pupil) of diameter 2 mm. The wavelength of light is given as 5000 Å (which is $5 \times 10^{-7}\,\text{m}$). The points are located 50 m away from the observer.
Step 2: Identify the Relevant Formula
According to the approximate resolution criterion for a circular aperture, the smallest angular separation $ \theta $ that can be resolved is given by:
$$
\theta \approx \frac{\lambda}{d},
$$
where:
$\lambda$ is the wavelength of light.
$d$ is the diameter of the aperture (here, the eye lens).
Once we have the angular separation $ \theta $, the corresponding linear separation $ y $ at a distance $ D $ is:
$$
y = \theta \times D.
$$
Step 3: Substitute the Known Values
Given:
$\lambda = 5 \times 10^{-7}\,\text{m}$
$d = 2 \times 10^{-3}\,\text{m}$
$D = 50\,\text{m}$
First, calculate the angular resolution:
$$
\theta = \frac{\lambda}{d} = \frac{5 \times 10^{-7}}{2 \times 10^{-3}}.
$$
Step 4: Calculate the Angular Resolution and Then the Linear Separation
1. Compute $ \theta $:
$$
\theta = \frac{5 \times 10^{-7}}{2 \times 10^{-3}}
= \frac{5 \times 10^{-7}}{2 \times 10^{-3}}
= 2.5 \times 10^{-4}.
$$
2. Now find $ y $ (the minimum separation) at distance $ D = 50\,\text{m}$:
$$
y = \theta \times D
= \left(2.5 \times 10^{-4}\right) \times 50
= 1.25 \times 10^{-2}\,\text{m}.
$$
Convert this to centimeters:
$$
1.25 \times 10^{-2}\,\text{m} = 1.25\,\text{cm}.
$$
Step 5: State the Final Answer
The minimum distance between two points that can be resolved by the eye under the given conditions is 1.25 cm.
