A single-slit diffraction pattern is formed by monochromatic electromagnetic radiation from a distant source passing through a slit 0.105 mm wide. At the point in the pattern 3.25° from the center of the central maximum, the total phase difference between wavelets from the top and bottom of the slit is 56.0 rad. (a) What is the wavelength of the radiation?

Verified step by step guidance

1. Identify the given values: slit width (a) = 0.105 mm, angle (θ) = 3.25°, and total phase difference (Δφ) = 56.0 rad.
2. Convert the slit width from millimeters to meters for consistency in SI units.
3. Understand that the phase difference (Δφ) between wavelets from the top and bottom of the slit can be expressed as Δφ = (2π/λ) * a * sin(θ), where λ is the wavelength of the radiation.
4. Rearrange the formula to solve for the wavelength λ: λ = (2π * a * sin(θ)) / Δφ.
5. Substitute the values of a, θ, and Δφ into the rearranged formula to calculate the wavelength λ.

Key Concepts

Diffraction

Diffraction is the bending of waves around obstacles and the spreading of waves when they pass through narrow openings. In the context of a single-slit diffraction pattern, it describes how light waves interfere with each other after passing through a slit, leading to a pattern of bright and dark fringes on a screen. The extent of diffraction depends on the wavelength of the light and the width of the slit.

Phase Difference

Phase difference refers to the difference in the phase of two waves at a given point in time. In diffraction, the phase difference between wavelets emanating from different parts of the slit affects the interference pattern observed. It is crucial for determining the conditions under which constructive or destructive interference occurs, which ultimately shapes the diffraction pattern.

Wavelength

Wavelength is the distance between successive peaks of a wave and is a fundamental property of electromagnetic radiation. It is directly related to the energy and frequency of the radiation, with shorter wavelengths corresponding to higher energy. In diffraction problems, the wavelength is essential for calculating the positions of the interference fringes and understanding the behavior of light as it interacts with slits.