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Ch 36: Diffraction
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
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Capitolo 36, Problema 2

Parallel rays of green mercury light with a wavelength of 546 nm pass through a slit covering a lens with a focal length of 60.0 cm. In the focal plane of the lens, the distance from the central maximum to the first minimum is 8.65 mm. What is the width of the slit?

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The problem involves single-slit diffraction, where the angular position of the first minimum is determined by the condition: a sin(θ) = mλ, where a is the slit width, λ is the wavelength of light, m is the order of the minimum (for the first minimum, m = 1), and θ is the diffraction angle.
The small-angle approximation can be applied here because the distance to the screen (focal length of the lens) is much larger than the distance from the central maximum to the first minimum. Using this approximation, sin(θ) ≈ tan(θ) ≈ y / f, where y is the distance from the central maximum to the first minimum (8.65 mm) and f is the focal length of the lens (60.0 cm).
Substitute sin(θ) into the diffraction condition: a (y / f) = λ. Rearrange this equation to solve for the slit width a: a = λ f / y.
Convert all quantities to consistent units before substituting into the formula. The wavelength λ is given as 546 nm, which should be converted to meters: λ = 546 × 10-9 m. The focal length f is 60.0 cm, which should be converted to meters: f = 0.600 m. The distance y is 8.65 mm, which should be converted to meters: y = 0.00865 m.
Substitute the converted values of λ, f, and y into the formula a = λ f / y to calculate the slit width a. This will give the final result for the width of the slit.

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Diffraction

Diffraction is the bending of waves around obstacles and the spreading of waves when they pass through narrow openings. In the context of light passing through a slit, diffraction patterns are formed, characterized by alternating bright and dark regions. The width of the slit influences the extent of diffraction, with narrower slits causing more pronounced spreading of the light waves.
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Diffraction

Single-Slit Diffraction Formula

The single-slit diffraction pattern can be quantitatively described using the formula for the position of minima: a sin(θ) = mλ, where 'a' is the slit width, 'θ' is the angle of the minima, 'm' is the order of the minimum, and 'λ' is the wavelength of the light. This relationship allows us to calculate the slit width when the positions of the minima are known, as in the given problem.
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Single Slit Diffraciton

Focal Length and Lens Behavior

The focal length of a lens is the distance from the lens to the point where parallel rays of light converge. In this scenario, the lens focuses the diffracted light from the slit, creating a pattern in the focal plane. Understanding how the focal length relates to the geometry of the setup is crucial for determining the positions of the diffraction minima and ultimately calculating the slit width.
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Lens Maker Equation
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