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. What is the intensity at this point, if the intensity at the center of the central maximum is I₀?

Verified step by step guidance

1. Understand the problem: We are dealing with a single-slit diffraction pattern. The intensity at a given angle θ from the central maximum is determined by the interference of wavelets originating from different parts of the slit. The phase difference between the wavelets is given, and we need to calculate the intensity at this angle relative to the central maximum intensity I₀.
2. Recall the formula for the intensity in a single-slit diffraction pattern: \( I(\theta) = I_0 \left( \frac{\sin(\beta)}{\beta} \right)^2 \), where \( \beta = \frac{\pi a \sin(\theta)}{\lambda} \). Here, \( a \) is the slit width, \( \lambda \) is the wavelength of the light, and \( \theta \) is the angle from the central maximum.
3. Relate the given phase difference \( \Delta \phi \) to the parameter \( \beta \). The phase difference \( \Delta \phi \) is related to \( \beta \) by \( \Delta \phi = 2\beta \). Using the given \( \Delta \phi = 56.0 \) rad, calculate \( \beta \) as \( \beta = \frac{\Delta \phi}{2} \).
4. Substitute \( \beta \) into the intensity formula. Compute \( \sin(\beta) \) and divide it by \( \beta \). Then square the result to find the relative intensity \( \frac{I(\theta)}{I_0} = \left( \frac{\sin(\beta)}{\beta} \right)^2 \).
5. Interpret the result: The value of \( \frac{I(\theta)}{I_0} \) represents the fraction of the central maximum intensity at the given angle. Multiply this fraction by \( I_0 \) to find the intensity \( I(\theta) \) at the specified angle.

Key Concepts

Single-Slit Diffraction

Single-slit diffraction occurs when waves pass through a narrow opening, causing them to spread out and create a pattern of light and dark fringes on a screen. The width of the slit and the wavelength of the light determine the characteristics of the diffraction pattern, including the angular position of the minima and maxima.

Phase Difference

Phase difference refers to the difference in the phase of two wavefronts at a given point in space. In the context of diffraction, it is crucial for determining how wavelets from different parts of the slit interfere with each other, affecting the intensity of the resulting pattern. A phase difference of 56.0 rad indicates how much the waves are out of sync at a specific angle.

Intensity of Light

The intensity of light is a measure of the power per unit area carried by a wave, often represented as I. In diffraction patterns, the intensity at any point can be calculated using the central maximum intensity (I0) and the phase difference, which influences constructive and destructive interference. The intensity decreases as one moves away from the central maximum due to the varying contributions of the wavelets.