A laser with a wavelength of 551 nm illuminates two narrow slits. The interference pattern from the double slits is viewed on a screen that is 1.50 m away. Over a distance of 46.0 mm there are 13.0 bright fringes with the first and last fringe situated exactly at each end of that distance. What is the spacing between the two slits?
Question
A laser with a wavelength of 551 nm illuminates two narrow slits. The interference pattern from the double slits is viewed on a screen that is 1.50 m away. Over a distance of 46.0 mm there are 13.0 bright fringes with the first and last fringe situated exactly at each end of that distance. What is the spacing between the two slits?
Solution
To solve this problem, we can use the formula for double-slit interference:
dsin(θ) = mλ
where:
- d is the spacing between the slits,
- θ is the angle between the central maximum and the mth order bright fringe,
- m is the order of the fringe, and
- λ is the wavelength of the light.
First, we need to find the angle θ. We can use the small angle approximation (sin(θ) ≈ tan(θ) ≈ θ for small θ), and the fact that tan(θ) = y/L, where y is the distance from the central maximum to the mth order fringe, and L is the distance from the slits to the screen.
From the problem, we know that there are 13 fringes over a distance of 46.0 mm, so the distance from the central maximum to the 13th order fringe is y = 46.0 mm / 2 = 23.0 mm = 23.0 * 10^-3 m. The distance from the slits to the screen is L = 1.50 m. So, tan(θ) = y/L = (23.0 * 10^-3 m) / (1.50 m) = 0.01533.
Next, we can find the order of the fringe. Since there are 13 fringes over the distance y, and the first and last fringe are at each end of that distance, the order of the fringe is m = 13 - 1 = 12.
Finally, we can substitute these values into the formula for double-slit interference to find the spacing between the slits:
d = m*λ / sin(θ) = 12 * (551 * 10^-9 m) / 0.01533 = 4.30 * 10^-4 m = 0.430 mm.
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