To find the distance on the screen for the first-order bright fringes created by two different wavelengths of coherent light passing through narrow slits, we can use the principles of interference in wave optics. The formula we will use is derived from the double-slit interference pattern, which states that the position of the bright fringes on the screen can be calculated using the equation:
Understanding the Interference Pattern
The position of the m-th order bright fringe (where m is an integer representing the order of the fringe) is given by:
y = (m * λ * L) / d
Where:
- y = distance from the central maximum to the m-th order bright fringe on the screen
- m = order of the fringe (1 for first-order)
- λ = wavelength of the light
- L = distance from the slits to the screen
- d = distance between the slits
Given Values
For this problem, we have:
- Wavelengths: λ₁ = 600 nm = 600 x 10-9 m and λ₂ = 470 nm = 470 x 10-9 m
- Distance between slits: d = 0.3 mm = 0.3 x 10-3 m
- Distance to the screen: L = 5 m
- Order of fringe: m = 1 (for first-order)
Calculating the Positions
Now, let's calculate the positions of the first-order bright fringes for both wavelengths.
For λ₁ = 600 nm
Substituting the values into the formula:
y₁ = (1 * 600 x 10-9 m * 5 m) / (0.3 x 10-3 m)
Calculating this gives:
y₁ = (600 x 5) / 0.3 = 10000 x 10-9 / 0.3 = 0.03333 m = 33.33 mm
For λ₂ = 470 nm
Now, substituting the second wavelength:
y₂ = (1 * 470 x 10-9 m * 5 m) / (0.3 x 10-3 m)
Calculating this gives:
y₂ = (470 x 5) / 0.3 = 7850 x 10-9 / 0.3 = 0.02617 m = 26.17 mm
Final Results
To summarize, the distances from the central maximum to the first-order bright fringes on the screen are:
- For the wavelength of 600 nm: 33.33 mm
- For the wavelength of 470 nm: 26.17 mm
This analysis shows how different wavelengths of light create distinct interference patterns, leading to varying positions of the bright fringes on the screen. Understanding this concept is crucial in optics and has applications in various fields, including physics and engineering.
