A student performs a double-slit experiment using a monochromatic light source with a wavelength of 5.00x10⁻⁷ m. The pattern appears on a screen 150 cm away ad the bright fringes are 0.40 cm apart. If the wavelength of the light used is changed to 7.50x10⁻⁷ m, what would the average distance between bright fringes become?
To find the average distance between bright fringes, we can use the formula for double-slit interference:
d * sin(θ) = m * λ
where:
- d is the distance between the slits,
- θ is the angle of the fringe,
- m is the order of the bright fringe,
- and λ is the wavelength of the light.
Given that the first wavelength used is 5.00x10⁻⁷ m and the distance between bright fringes is 0.40 cm, we can calculate the distance between the slits (d):
d = (m * λ) / sin(θ)
We need two sets of values to compare in order to find the average distance between bright fringes. Let's say that for the second wavelength of 7.50x10⁻⁷ m, the new distance between bright fringes is x cm. Now, we can write another equation using the same formula:
d * sin(θ) = m * λ'
where:
- λ' is the new wavelength of the light.
So, for the new wavelength, the distance between the slits (d') can be calculated as:
d' = (m * λ') / sin(θ)
To find the difference in distance between bright fringes, we subtract:
Δx = d' - d
Next, we can substitute the expressions for d and d' in terms of λ and λ' respectively:
Δx = [(m * λ') / sin(θ)] - [(m * λ) / sin(θ)]
Since the distance between bright fringes is given by 0.40 cm, we can substitute this value in for Δx:
0.40 cm = [(m * λ') / sin(θ)] - [(m * λ) / sin(θ)]
Now, we have an equation that relates the two wavelengths and the distances between bright fringes. Given that the first wavelength is 5.00x10⁻⁷ m and the second wavelength is 7.50x10⁻⁷ m, we can solve for the unknown average distance between bright fringes (x):
0.40 cm = [(m * 7.50x10⁻⁷ m) / sin(θ)] - [(m * 5.00x10⁻⁷ m) / sin(θ)]