Problem 2: Two parallel slits are illuminated by two wavelengths one of which is 5.8 × 10-7 m. On the screen the fourth dark line of the known wavelength coincides with the fifth bright line of the light of the unknown wavelength. Find the unknown wavelength.

Solution

Assumption Let the known wavelength is denoted by λ_{1} and the unknown by λ_{2}.

Given data
λ_{1} = 5.8 × 10^{-7} m

Required
λ_{2} = ?

Formula

where m = 0, 1, 2, …

m = 1, 2, 3, …

(Noting that for 4^{th} dark line of the first wavelength, m = 3, and for fifth bright line of second wavelength, m = 5. WHY?)

Strategy: We apply formula (1) to find the distance of the 4^{th} dark line from the center of the screen. Since the 5^{th} bright fringe of the second wavelength is coinciding this dark fringe of the first wavelength, therefore, it will also be the distance of the 5^{th} bright fringe of the second wavelength. Then we use formula (2) to find the second wavelength. Then we equate both equations and fine the second wavelength.

So to find Y we use formula (1).

Similarly, for the second given wavelength,

Now according to the condition of the question the dark line of equation (3) coincides with the bright line of equation (4). Therefore, equating

Cancelling common factors d and L on both sides and simplifying,

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