A slit 0.240 mm wide is illuminated by parallel light rays of wavelength 540 nm. The diffraction pattern is observed on a screen that is 3.00 m from the slit. The intensity at the center of the central maximum (u = 0°) is 6.00 x 10^-6 W/m2. (b) What is the intensity at a point on the screen midway between the center of the central maximum and the first minimum?

Verified step by step guidance

1. First, understand the problem involves single-slit diffraction, where light of a specific wavelength passes through a narrow slit and forms a diffraction pattern on a screen. The intensity of the light varies across the pattern.
2. Calculate the angle \( \theta \) for the first minimum using the formula for minima in single-slit diffraction: \( a \sin(\theta) = m\lambda \), where \( a \) is the slit width, \( \lambda \) is the wavelength, and \( m \) is the order of the minimum (\( m = 1 \) for the first minimum).
3. Determine the angle \( \theta \) for the point midway between the center of the central maximum and the first minimum. This is half the angle calculated for the first minimum.
4. Use the formula for the intensity distribution in single-slit diffraction, \( I(\theta) = I_0 \left( \frac{\sin(\beta)}{\beta} \right)^2 \), where \( \beta = \frac{\pi a \sin(\theta)}{\lambda} \) and \( I_0 \) is the intensity at the center of the central maximum.
5. Substitute the values of \( \theta \), \( a \), \( \lambda \), and \( I_0 \) into the intensity formula to find the intensity at the specified point on the screen.

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 light, diffraction patterns arise when light encounters a slit, leading to the formation of maxima and minima on a screen. The extent of diffraction depends on the wavelength of the light and the width of the slit, with narrower slits causing more pronounced diffraction effects.

Intensity of Light

The intensity of light is defined as the power per unit area carried by a wave, typically measured in watts per square meter (W/m²). In diffraction patterns, intensity varies with position on the screen, with the central maximum being the brightest point. The intensity decreases as one moves away from the center, reaching zero at the minima, where destructive interference occurs.

Interference Patterns

Interference patterns result from the superposition of waves, leading to regions of constructive and destructive interference. In a single-slit diffraction experiment, light waves emanating from different parts of the slit interfere with each other, creating a series of bright and dark fringes on the screen. The position of these fringes can be calculated using the wavelength of the light and the geometry of the slit.