(a) How far away can a human eye distinguish two car headlights 2.0 m apart? Consider only diffraction effects and assume an eye pupil diameter of 6.0 mm and a wavelength of 560 nm. (b) What is the minimum angular separation an eye could resolve when viewing two stars, considering only diffraction effects? In reality, it is about of arc. Why is it not equal to your answer in (b)?

Verified step by step guidance

1. Step 1: Understand the problem and identify the key concepts. This problem involves the concept of diffraction and the resolving power of the human eye. The Rayleigh criterion is used to determine the minimum angular separation that can be resolved. The formula for the angular resolution is θ = 1.22 * (λ / D), where λ is the wavelength of light and D is the diameter of the aperture (in this case, the pupil of the eye).
2. Step 2: Calculate the angular resolution for part (a). Use the given values: λ = 560 nm = 560 × 10⁻⁹ m and D = 6.0 mm = 6.0 × 10⁻³ m. Substitute these values into the formula θ = 1.22 * (λ / D) to find the angular resolution in radians.
3. Step 3: Relate the angular resolution to the linear distance for part (a). The angular resolution θ is related to the linear separation of the headlights (s) and the distance to the car (L) by the small-angle approximation: θ ≈ s / L. Rearrange this equation to solve for L: L = s / θ. Use s = 2.0 m and the θ value calculated in Step 2 to find the maximum distance at which the two headlights can be resolved.
4. Step 4: For part (b), calculate the minimum angular separation for two stars. Use the same formula θ = 1.22 * (λ / D) with the same values for λ and D. This will give the theoretical minimum angular separation that the human eye can resolve when viewing stars.
5. Step 5: Address the discrepancy between the theoretical and real-world resolving power. In reality, the resolving power of the human eye is affected by factors such as imperfections in the eye's lens, atmospheric turbulence, and other biological and environmental factors. These effects reduce the actual resolving power compared to the idealized diffraction-limited value calculated in part (b).

Key Concepts

Diffraction

Diffraction is the bending of waves around obstacles and the spreading of waves when they pass through small openings. In optics, it is particularly significant when dealing with light waves, as it limits the resolution of optical systems. The extent of diffraction depends on the wavelength of light and the size of the aperture, which in this case is the pupil of the human eye.

Rayleigh Criterion

The Rayleigh Criterion is a formula used to determine the minimum angular separation at which two point sources can be resolved. According to this criterion, two sources are considered resolvable when the central maximum of one diffraction pattern coincides with the first minimum of the other. This principle is crucial for understanding how the human eye perceives two closely spaced light sources, such as car headlights or stars.

Angular Resolution

Angular resolution refers to the smallest angle between two objects that an optical system can distinguish. It is influenced by factors such as the diameter of the aperture and the wavelength of light. In the context of the human eye, angular resolution determines how well we can differentiate between two closely spaced light sources, which is essential for tasks like identifying distant stars or car headlights.