Two parallel slits are illuminated by light composed of two wavelengths. One wavelength is λA = 553nm. The other wavelength is λB and is unknown. On a viewing screen, the light with wavelength λA = 553nm produces its third-order bright fringe at the same place where the light with wavelength λB produces its fourth dark fringe. The fringes are counted relative to the central or zeroth-order bright fringe. What is the unknown wavelength?
λB = 442 nm
To solve this problem, we can use the formula for the position of bright fringes in a double-slit interference pattern:
y = (λ*L) / d
where:
y is the position of the fringe on the viewing screen,
λ is the wavelength of the light,
L is the distance from the slits to the screen,
and d is the distance between the slits.
Let's first consider the situation for the light with wavelength λA = 553nm:
yA = (λA * L) / d
Now, let's consider the situation for the light with the unknown wavelength λB:
yB = (λB * L) / d
Given that the third-order bright fringe for λA coincides with the fourth dark fringe for λB, we can set up the following equation:
3 * yA = (4 * yB)
Substituting the respective expressions for yA and yB:
3 * [(λA * L) / d] = 4 * [(λB * L) / d]
Simplifying the equation:
(λA * L) / d = (4/3) * (λB * L) / d
The distance L and the distance d are common to both sides of the equation and can be canceled out:
λA = (4/3) * λB
Solving for λB:
λB = (3/4) * λA
Substituting the known value of λA = 553nm:
λB = (3/4) * 553nm
Calculating the unknown wavelength λB:
λB = 415.75nm
Therefore, the unknown wavelength is 415.75nm.
To find the unknown wavelength, we can use the principles of interference and the formula for fringe separation.
First, let's assign variables to the known information:
λA = 553 nm
λB = unknown wavelength
m1 = order of the bright fringe for λA = 3
m2 = order of the dark fringe for λB = 4
Now, let's establish a relationship between the two wavelengths and their corresponding fringes. For constructive interference (bright fringes), the path difference between the two slits must be an integer multiple of the wavelength. For destructive interference (dark fringes), the path difference must be an odd multiple of half-wavelength.
For λA:
m1λA = path difference
For λB:
(m2 + 0.5)λB = path difference
Since the third-order bright fringe for λA coincides with the fourth-order dark fringe for λB at the same position, the path differences will be the same.
Therefore, we can equate the two expressions:
m1λA = (m2 + 0.5)λB
Plugging in the known values:
3 * 553 nm = (4 + 0.5) * λB
Simplifying the equation:
1659 nm = 4.5 * λB
Dividing both sides by 4.5:
λB = 1659 nm / 4.5
Calculating the value:
λB ≈ 368 nm
Therefore, the unknown wavelength (λB) is approximately 368 nm.