White light is used to illuminate the two slits in a Young's double slit experiment. The separation between the slits is b and the screen is at a distance d`(gtb) from the slits. At a point on the screen directly in front of one of the slits, certain wavelength are missing. Some of these missing wavelength are

Options

A. `lambda = (b^(2))/(d)`
B. `lambda = (2b^(2))/(d)`
C. `lambda = (b^(2))/(3d)`
D. `lambda = (2b^(2))/(3d)`

Answer

A, C

Explanation

From the figure : `S_(2)P^(2) =S_(1)P^(2) +S_(1)S_(2)^(2), S_(2)P^(2) - S_(2)P^(2) - S_(1)P^(2) =S_(1)S_(2)^(2)`
`(S_(2)P-S_(1)P) (S_(2)P+S_(1)P) = S_(1)S_(2)^(2)`
`(S_(2)P-S_(1)P) = (S_(1)S_(2)^(2))/((S_(2)P+S_(1)P)) = (b^(2))/(2d) [:' S_(2)P = S_(1)P=d]`
Missing wavelengths means dark fringes, hence path difference `(x) = (2n -1) (lambda)/(2) = (b^(2))/(2d) lambda = (b^(2))/((2n - 1)d)`
For `n = 1 implies lambda_(1) = (b^(2))/(d) Fro n = 2 implies lambda_(2) = (b^(2))/(3d)`
For `n = 3 implies lambda_(3) = (b^(2))/(5d)` and so on.