A slit 0.240 mm wide is illuminated by parallel light rays...

A slit 0.240 $\mathrm{mm}$ wide is illuminated by parallel light rays of wavelength 540 $\mathrm{nm}$ . The diffraction pattern is observed on a screen that is 3.00 $\mathrm{m}$ from the slit. The intensity at the center of the central maximum $\left(\theta=0^{\circ}\right)$ is $6.00 \times 10^{-6} \mathrm{W} / \mathrm{m}^{2} .$ (a) What is the distance on the screen from the center of the central maximum to the first minimum? (b) What is the intensity at a point on the screen midway between the center of the central maximum and the first minimum?

Key Concepts

Single-Slit Diffraction

Single-slit diffraction is a phenomenon that occurs when coherent light passes through a narrow opening, causing the light to spread out and form an interference pattern on a screen. This pattern of alternating bright and dark regions is due to the constructive and destructive interference of light waves emanating from different parts of the slit. The overall shape of the diffraction pattern is characterized by a central maximum that is much brighter and wider than the secondary maxima.

Diffraction Minima Condition

The positions of the minima (dark fringes) in a single-slit diffraction pattern are determined by the condition a sin(?) = m?, where 'a' is the width of the slit, '?' is the angle of diffraction, '?' is the wavelength of the light, and 'm' is an integer (excluding m = 0). This equation indicates the angles at which the light waves cancel each other out, leading to dark bands on the observation screen.

Small Angle Approximation

The small angle approximation is often used in diffraction problems when the angles involved are small. In such cases, sin(?) and tan(?) can be approximated by ? in radians. This simplification allows for straightforward calculation of distances on the screen from the diffraction angles by relating ? directly to the distance from the center using the relation y ? L?, where L is the distance from the slit to the screen.

Intensity Distribution in Diffraction

The intensity distribution in a single-slit diffraction pattern is described by an equation involving the square of the sinc function, typically written as I(?) = I(0)[sin(?)/?]^2, where ? = (?a sin(?))/?. This function shows how the intensity decreases from a maximum at the center (? = 0) to minima at the angles determined by the diffraction condition. Evaluating this expression at different angles allows one to calculate the relative brightness at various points in the diffraction pattern.

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