When driving at night, your eyes’ pupils have dilated to a 7.5-mm diameter. If your vision is diffraction limited, what would be the greatest distance at which you could resolve the two headlights of an oncoming car, which are spaced 1.5 m apart? Assume a wavelength of 550 nm for the light.

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Start by identifying the key formula for diffraction-limited resolution: the angular resolution θ is given by 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).

Convert the given values into consistent SI units: λ = 550 nm = 550 × 10⁻⁹ m, and D = 7.5 mm = 7.5 × 10⁻³ m.

Calculate the angular resolution θ using the formula θ = 1.22 * (λ / D). This will give the smallest angular separation that can be resolved by the eye.

Relate the angular resolution θ to the physical separation of the headlights and the distance to the car using the small-angle approximation: θ ≈ s / L, where s is the separation between the headlights (1.5 m) and L is the distance to the car. Rearrange this to solve for L: L = s / θ.

Substitute the calculated value of θ and the given value of s into the equation L = s / θ to find the maximum distance at which the two headlights can be resolved. Ensure all units are consistent during substitution.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Diffraction Limit

The diffraction limit refers to the fundamental limit on the resolution of optical systems due to the wave nature of light. When light passes through an aperture, such as the pupil of the eye, it spreads out, causing a limit to how closely two points can be resolved. This phenomenon is described by the Rayleigh criterion, which states that two point sources are resolvable when the central maximum of one diffraction pattern coincides with the first minimum of another.

Rayleigh Criterion

The Rayleigh criterion provides a formula to determine the minimum angular separation at which two point sources can be distinguished. It is given by θ = 1.22(λ/D), where θ is the angular resolution in radians, λ is the wavelength of light, and D is the diameter of the aperture. This criterion is crucial for understanding how the size of the pupil affects the ability to resolve objects, especially in low-light conditions.

Resolving Distance

Resolving distance is the maximum distance at which two objects can be distinguished as separate entities. It is influenced by the angular resolution and the physical separation of the objects. In the context of the question, it involves calculating how far away the headlights of an oncoming car can be while still being resolvable, based on the pupil size and the wavelength of light, using the principles of diffraction and the Rayleigh criterion.