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.
Assumption Let the known wavelength is denoted by λ1 and the unknown by λ2.
λ1 = 5.8 × 10-7 m
λ2 = ?
where m = 0, 1, 2, …
m = 1, 2, 3, …
(Noting that for 4th 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 4th dark line from the center of the screen. Since the 5th bright fringe of the second wavelength is coinciding this dark fringe of the first wavelength, therefore, it will also be the distance of the 5th 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,