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. (a) What is the distance on the screen from the center of the central maximum to the first minimum?

Verified step by step guidance

1. First, understand that the problem involves single-slit diffraction, where light passing through a narrow slit spreads out and creates a pattern of bright and dark regions on a screen.
2. Use the formula for the position of the first minimum in a single-slit diffraction pattern, which is given by \( y = L \tan(\theta) \), where \( L \) is the distance from the slit to the screen, and \( \theta \) is the angle at which the first minimum occurs.
3. Calculate the angle \( \theta \) for the first minimum using the formula \( \theta = \frac{m \lambda}{a} \), where \( m = 1 \) for the first minimum, \( \lambda \) is the wavelength of the light, and \( a \) is the width of the slit.
4. Substitute the values of \( L \), \( \lambda \), and \( a \) into the equations to find \( \theta \) and then use it to find \( y \), the distance from the center of the central maximum to the first minimum on the screen.
5. Since \( \theta \) is typically small in diffraction problems, you can approximate \( \tan(\theta) \approx \sin(\theta) \approx \theta \) in radians for simplifying calculations.

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 regions of constructive and destructive interference. The width of the slit and the wavelength of the light significantly influence the diffraction pattern observed on a screen.

Minima in Diffraction Patterns

In a single-slit diffraction pattern, minima occur at specific angles where destructive interference takes place. The position of the first minimum can be calculated using the formula a sin(θ) = mλ, where 'a' is the slit width, 'm' is the order of the minimum (m=1 for the first minimum), and 'λ' is the wavelength of the light. This relationship helps determine the distance from the central maximum to the first minimum on the observation screen.

Intensity of Light

The intensity of light is defined as the power per unit area and is measured in watts per square meter (W/m²). In diffraction patterns, the intensity varies with position due to the interference of light waves. The intensity at the central maximum is typically the highest, and it decreases as one moves towards the minima, which can be calculated using the intensity distribution formula derived from the diffraction pattern.