Two parallel slits are illuminated by light of two wavelengths, one of which is 5.8 x 10⁻⁷ m. On the screen the fourth dark line of the known wavelength coincides with the fifth bright line of the light of unknown wavelength. Find the unknown wavelength. (5.2 x 10⁻⁷ m Book Answer is wrong.) (Correct Ans. is 4.1 x 10⁻⁷ m)



Given:

Let the known Wavelength of dark fringe= λ₁ = 5.8 ｘ 10⁻⁷ m

For fourth dark fringes = m₁ = 3

For fifth bright fringes = m₂ = 5

As given 

`\ (Y_5)_{bright}` = `\ (Y_4)_{dark}`


To Find:

The unknown Wavelength of bright fringe = λ₂ =  ?


Solution:


For `5^{th}` bright fringe we have

`\ (Y_5)_{bright}` = m₂ L `\frac {λ₂}{d}`


and For `4^{th}` dark fringe we have

`\ (Y_4)_{dark}` = (m₁ + `\frac{1}{2}`) L `\frac {λ₁}{d}`

Now according to the given condition i.e.

`\ (Y_5)_{bright}` = `\ (Y_4)_{dark}`

Or

m₂ L `\frac {λ₂}{d}` = (m₁ + `\frac{1}{2}`) L `\frac {λ₁}{d}`

L and d will be canceled from both sides of equation we remain with

m₂ λ₂ = (m₁ + `\frac{1}{2}`) λ₁


now by putting the corresponding values

5 λ₂ = (3 + `\frac{1}{2}`) 5.8 ｘ 10⁻⁷ m

or

5 λ₂ = `\frac{7}{2}` ｘ 5.8 ｘ 10⁻⁷ m

or

5 λ₂ = 20.3 ｘ 10⁻⁷ m

or

λ₂ = `\frac{20.3 ｘ 10⁻⁷ m}{5}`

or

λ₂ = 4.06 ｘ 10⁻⁷ m

or

λ₂ = 4.1 ｘ 10⁻⁷ m -----------------Ans.



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