Final answer:
To find the separation between the positions of the first maxima for two different wavelengths, we calculate the position of each maximum using the formula y = (Lλ)/d and then find their difference. The separation for the 590 nm and 596 nm wavelengths is 45 μm.
Step-by-step explanation:
To calculate the separation between the positions of the first maxima for sodium light wavelengths of 590 nm and 596 nm in a diffraction experiment, we need to use the diffraction formula for a single slit: d sin(θ) = mλ, where d is the slit aperture, θ is the angle of the diffraction maximum, m is the order number of the maximum, and λ is the wavelength of the light.
In this case, the order of the maximum m is 1, since we are looking for the first maxima. The aperture width d is given as 2×10−4 m. The distance L from the slit to the screen is 1.5 m. The position y of a maximum on the screen is related to the angle θ through the relationship y = L tan(θ), but if the angle is small (which is generally true for diffraction patterns), we can approximate tan(θ) with sin(θ): y ≈ L sin(θ).
For the first maximum m = 1, the position is given by: y = (Lλ)/d. Thus, we compute the positions of the maxima for the two wavelengths separately and then find their separation as the difference between the two positions.
Using the provided wavelengths:
- For λ = 590 nm: y590 = (1.5 m × 590×10−9 m) /(2×10−4 m) = 4.425×10−3 m
- For λ = 596 nm: y596 = (1.5 m × 596×10−9 m) /(2×10−4 m) = 4.470×10−3 m
The separation between the two first maxima is the difference of these two positions:
Separation = |y596 - y590| = |4.470×10−3m - 4.425×10−3m| = 4.5×10−5m or 45 μm