A mixture of light, consisting of wavelength 590 nm and an unknown wavelength, illuminates Young's double slit and gives rise to two overlapping interference patterns on the screen. The central maximum of both lights coincide. Further, it is observed that the third bright fringe of known light coincides with the 4th bright fringe of the unknown light. From this data, the wavelength of the unknown light is:?

Let λ1 be the wavelength of the known light and λ2 be the wavelength of the unknown light.
λ1 = 590 nm
In Young's double slit experiment, the position of the nth bright fringe is given by:
xn = nλD/d
where:
xn is the position of the nth bright fringe
n is the order of the fringe
λ is the wavelength of light
D is the distance between the slits and the screen
d is the distance between the slits
Given that the third bright fringe of known light (λ1) coincides with the 4th bright fringe of unknown light (λ2), we have:
3λ1D/d = 4λ2D/d
The D/d terms cancel out:
3λ1 = 4λ2
λ2 = (3/4)λ1
λ2 = (3/4) × 590 nm
λ2 = 442.5 nm
Therefore, the wavelength of the unknown light is 442.5 nm.