Question

Think of the pupil of your eye as a circular aperture \(5.00 \mathrm{~mm}\) in diameter. Assume you are viewing light of wavelength \(550 . \mathrm{nm},\) to which your eyes are maximally sensitive. a) What is the minimum angular separation at which you can distinguish two stars? b) What is the maximum distance at which you can distinguish the two headlights of a car mounted \(1.50 \mathrm{~m}\) apart?

Short answer

Answer: The minimum angular separation between two stars that a person can distinguish is approximately 7.68e-3 degrees. The maximum distance at which a person can distinguish the two headlights of a car mounted 1.50 m apart is approximately 11,172.63 meters.

Step-by-step solution

Understand the Rayleigh criterion

According to the Rayleigh criterion, the minimum angular separation (θ) that can be resolved by a circular aperture is given by the formula: θ = 1.22 * (λ / D) where θ - Minimum angular separation in radians λ - Wavelength of the light (in this case, 550 nm) D - Diameter of the aperture (in this case, the pupil diameter: 5.00 mm)

Calculate the minimum angular separation between two stars

Using the given values for λ and D, we can calculate the minimum angular separation at which a person can distinguish two stars using the formula mentioned above: Convert wavelength to meters: λ = 550 nm * (1 m / 1e9 nm) = 550e-9 m Convert diameter to meters: D = 5 mm * (1 m / 1000 mm) = 5e-3 m Now, plug these values into the formula: θ = 1.22 * (550e-9 m / 5e-3 m) θ = 1.34e-4 radians. This is the minimum angular separation in radians. To convert to degrees, multiply by (180 / π): θ = 1.34e-4 * (180/π) ≈ 7.68e-3 degrees. Minimum angular separation between two stars is approximately 7.68e-3 degrees.

Calculate the maximum distance at which headlights can be distinguished

The angular separation (θ) between the two headlights mounted 1.50 m apart can be related to the distance (d) from an observer by the formula: θ = tan^(-1)(s / d), where s - Separation between the headlights (1.50m) θ - Minimum angular separation (1.34e-4 radians) First, solve for the distance (d) using the minimum angular separation: d = s / tan(θ) d = 1.50 m / tan(1.34e-4) Calculate the distance (d) d ≈ 11,172.63 m The maximum distance at which a person can distinguish the two headlights of a car mounted 1.50 m apart is approximately 11,172.63 meters.