Light from a helium-neon laser (λ = 633 nm) passes through a circular aperture and is observed on a screen 4.0 m behind the aperture. The width of the central maximum is 2.5 cm. What is the diameter (in mm) of the hole?

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Understand the problem: The central maximum in the diffraction pattern is caused by light passing through a circular aperture. The width of the central maximum is related to the aperture diameter by the diffraction formula for a circular aperture.

Use the formula for the angular width of the central maximum in a circular aperture: \( \theta = 1.22 \frac{\lambda}{D} \), where \( \lambda \) is the wavelength of light, \( D \) is the diameter of the aperture, and \( \theta \) is the angular width of the central maximum.

Relate the angular width \( \theta \) to the physical width \( w \) of the central maximum on the screen using the small-angle approximation: \( \theta \approx \frac{w}{2L} \), where \( w \) is the width of the central maximum (2.5 cm) and \( L \) is the distance to the screen (4.0 m).

Combine the two equations: \( \frac{w}{2L} = 1.22 \frac{\lambda}{D} \). Rearrange to solve for \( D \): \( D = 1.22 \frac{\lambda \cdot 2L}{w} \).

Substitute the known values into the equation: \( \lambda = 633 \text{ nm} = 633 \times 10^{-9} \text{ m} \), \( L = 4.0 \text{ m} \), and \( w = 2.5 \text{ cm} = 0.025 \text{ m} \). Simplify the expression to find \( D \) in meters, then convert to millimeters.

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주요 개념

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Diffraction

Diffraction is the bending of waves around obstacles and the spreading of waves when they pass through small openings. In the context of light, diffraction patterns are created when coherent light, such as that from a laser, encounters an aperture. The resulting pattern on a screen can be analyzed to determine properties of the aperture, such as its size.

Central Maximum

The central maximum is the brightest and widest part of the diffraction pattern produced when light passes through an aperture. It occurs directly in line with the light source and is flanked by successive dark and bright fringes. The width of the central maximum is influenced by the wavelength of the light and the size of the aperture, making it a key factor in determining the aperture's diameter.

Wavelength and Aperture Size Relationship

The relationship between the wavelength of light and the size of the aperture is crucial in diffraction phenomena. According to the principles of wave optics, smaller apertures relative to the wavelength produce more pronounced diffraction effects. This relationship can be quantitatively described using formulas that relate the width of the central maximum to the wavelength and the diameter of the aperture, allowing for calculations of the aperture size.