Parallel rays of monochromatic light with wavelength 568 nm illuminate two identical slits and produce an interference pattern on a screen that is 75.0 cm from the slits. The centers of the slits are 0.640 mm apart and the width of each slit is 0.434 mm. If the intensity at the center of the central maximum is 5.00x10^-4 W/m2, what is the intensity at a point on the screen that is 0.900 mm from the center of the central maximum?

Verified step by step guidance

1. Identify the given values: wavelength (λ) = 568 nm, distance to screen (D) = 75.0 cm, distance between slits (d) = 0.640 mm, width of each slit (a) = 0.434 mm, and the intensity at the central maximum (I_0) = 5.00x10^-4 W/m^2.
2. Convert all measurements to meters for consistency in units. For example, convert the wavelength from nm to meters by multiplying by 1x10^-9, and distances from mm or cm to meters by multiplying by 1x10^-3 or 1x10^-2, respectively.
3. Calculate the angle θ for the point 0.900 mm from the center of the central maximum using the small angle approximation, θ ≈ x/D, where x is the distance from the center of the central maximum.
4. Use the double slit interference formula to find the intensity at this point: I = I_0 \left(\frac{\sin(\beta)}{\beta}\right)^2 \left(\frac{\sin(N\alpha)}{N\sin(\alpha)}\right)^2, where \beta = \frac{\pi a \sin(\theta)}{\lambda} and \alpha = \frac{\pi d \sin(\theta)}{\lambda}.
5. Substitute the values of θ, λ, a, d, and I_0 into the formula to calculate the intensity at the point 0.900 mm from the center of the central maximum.

Key Concepts

Interference of Light

Interference occurs when two or more coherent light waves overlap, resulting in a new wave pattern. In the context of double slits, constructive interference leads to bright fringes, while destructive interference results in dark fringes. The positions of these fringes depend on the wavelength of the light and the geometry of the slits, which is crucial for analyzing the intensity at various points on the screen.

Intensity of Light

Intensity is defined as the power per unit area carried by a wave, typically measured in watts per square meter (W/m²). In interference patterns, the intensity at a point on the screen varies due to the superposition of light waves from the slits. The intensity can be calculated using the amplitude of the waves and the interference conditions, which is essential for determining the intensity at specific locations.

Young's Double Slit Experiment

Young's Double Slit Experiment demonstrates the wave nature of light through the creation of an interference pattern. The distance between the slits, the wavelength of light, and the distance to the screen are key parameters that influence the pattern. This experiment provides a framework for calculating the positions and intensities of the interference fringes, which is necessary for solving the given problem.