In a two-slit experiment using coherent light, the distance between the slits and the screen is 1.10 m, and the distance between the slits is 0.100 mm. If the first-order bright fringe is measured to be 3.40 cm from the centerline, what is the wavelength of the light?
Question
In a two-slit experiment using coherent light, the distance between the slits and the screen is 1.10 m, and the distance between the slits is 0.100 mm. If the first-order bright fringe is measured to be 3.40 cm from the centerline, what is the wavelength of the light?
Solution
To solve this problem, we can use the formula for the double-slit interference pattern:
dsin(θ) = mλ
where:
- d is the distance between the slits,
- θ is the angle between the centerline and the position of the fringe,
- m is the order of the fringe, and
- λ is the wavelength of the light.
First, we need to calculate the angle θ. We can use the small angle approximation (sin(θ) ≈ tan(θ) ≈ θ for small θ), which is valid because the distance to the screen is much larger than the distance between the fringes.
The tangent of the angle is the ratio of the opposite side (the distance from the centerline to the fringe, which is 3.40 cm or 0.034 m) to the adjacent side (the distance from the slits to the screen, which is 1.10 m). So,
tan(θ) = 0.034 m / 1.10 m = 0.0309
Since the angle is small, we can say that θ ≈ 0.0309.
Next, we can substitute the known values into the formula:
0.100 mm * 0.0309 = 1 * λ
We need to convert the distance between the slits from mm to m, so d = 0.100 mm = 0.0001 m. Then,
0.0001 m * 0.0309 = λ
Finally, we solve for λ:
λ = 0.0001 m * 0.0309 = 3.09 * 10^-6 m = 309 nm
So, the wavelength of the light is approximately 309 nm.
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