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. (b) 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. First, understand that the intensity distribution in a single-slit diffraction pattern can be described using the formula $I = I_0 \left(\frac{\sin(\beta)}{\beta}\right)^2$, where $\beta = \frac{\pi a \sin(\theta)}{\lambda}$, $I_0$ is the maximum intensity at the center, $a$ is the slit width, $\theta$ is the angle from the central maximum, and $\lambda$ is the wavelength of the light.
2. Calculate the wavelength $\lambda$ using the given phase difference $\Delta \phi = 56.0 \text{ rad}$ and the relationship $\Delta \phi = \frac{2\pi a \sin(\theta)}{\lambda}$. Rearrange this formula to solve for $\lambda$.
3. Substitute the value of $\lambda$ back into the expression for $\beta = \frac{\pi a \sin(\theta)}{\lambda}$ to find the value of $\beta$.
4. Compute the value of $\sin(\beta)/\beta$ using the value of $\beta$ calculated in the previous step.
5. Finally, substitute the value of $\sin(\beta)/\beta$ into the intensity formula $I = I_0 \left(\frac{\sin(\beta)}{\beta}\right)^2$ to find the relative intensity $I$ at the angle $\theta = 3.25^\circ$ compared to the central maximum intensity $I_0$.

Key Concepts

Single-Slit Diffraction

Single-slit diffraction occurs when waves pass through a narrow opening and spread out, creating a pattern of light and dark fringes on a screen. The width of the slit and the wavelength of the light determine the pattern's characteristics, including the angular position of the minima and maxima. This phenomenon illustrates the wave nature of light and is described mathematically by the diffraction formula, which relates the slit width, wavelength, and angle of observation.

Phase Difference

Phase difference refers to the difference in the phase of two waves at a given point in time and space. In the context of diffraction, it is crucial for determining how waves interfere with each other, leading to constructive or destructive interference. The total phase difference between wavelets from different parts of the slit affects the intensity of the resulting diffraction pattern, with specific angles corresponding to minima and maxima in intensity.

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 varies with position due to the interference of light waves. The intensity at any point can be calculated relative to the maximum intensity (I0) at the center of the pattern, using the formula that incorporates the phase difference and the slit width, which helps predict the brightness of the fringes observed.