Ch. 35 - Diffraction

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 35 - Diffraction

Problem 32

Chapter 34, Problem 32

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.

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Related Practice

Textbook Question

(a) Derive an expression for the intensity in the interference pattern for three equally spaced slits. Express in terms of δ = 2πd sin θ / λ where d is the distance between adjacent slits and assume the slit width D ≈ λ.

(b) Show that there is only one secondary maximum between principal peaks.

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Textbook Question

(III) Derive an expression for the intensity in the interference pattern for three equally spaced slits. Express in terms of δ = 2πd sin θ / λ where d is the distance between adjacent slits and assume the slit width D ≈ λ . Show that there is only one secondary maximum between principal peaks.

Textbook Question

The nearest neighboring star to the Sun is about 4 light-years away. If a planet happened to be orbiting this star at an orbital radius equal to that of the Earth–Sun distance, what minimum diameter would an Earth-based telescope’s aperture have to be in order to obtain an image that resolved this star–planet system? Assume the light emitted by the star and planet has a wavelength of 550 nm.

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Textbook Question

A diffraction grating has 6.5 x 10⁵ slits/m. Find the angular spread in the second-order spectrum between red light of wavelength 7.0 x 10⁻⁷ m and blue light of wavelength 4.5 x 10⁻⁷ m.

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Textbook Question

A 3800-slit/cm grating produces a third-order fringe at a 35.0° angle. What wavelength of light is being used?

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Textbook Question

Show that the second- and third-order spectra of white light produced by a diffraction grating always overlap. What wavelengths overlap?