Question

Two parallel slits are illuminated by light composed of two wavelengths. Wavelength A is 564 nm and the other is wavelength B and is unknown. On a viewing screen, the light of wavelength A produces a third-order bright fringe at the same place where the light with wavelength B produces its fourth dark fringe. The fringes are counted relative to the central or zeroth-order bright fringe. What is the unknown wavelength?

Answer 1

The unknown wavelength is 376 nm

Step-by-step explanation:

We are given two parallel slits that are illuminated by light composed of two wavelengths, λa = 564 nm and the other λb which is unknown.

On a viewing screen, the light whose wavelength is known produces its third dark fringe at the same place where the light with wavelength B produces its fourth dark fringe.

For the wavelength A: (bright fringes)

sinθ = mλa/d

where m = 3

sinθ = 3λa/d

For the wavelength B: (dark fringes)

sinθ = (m + ½)λb/d

where m = 4

sinθ = (4 + ½)λb/d

Since we both fringes are produced at the same place,

3λa/d = (4 + ½)λb/d

d cancels out

3λa = (4 + ½)λb

λb = 3λa/(4 + ½)

λb = 3(564)/(4.5)

λb = 376 nm

Therefore, the other wavelength is 376 nm.

Answer 2

423nm

Step-by-step explanation:

To find the unknown wavelength you take into account the distance y to the maximum central fringe, for light fringes and dark fringes.

- for light fringes:

`dsin\theta=m\lambda\\\\sin\theta\approx\theta=(y)/(D)\\\\y=(m\lambda_1D)/(d)`

- for dark fringes:

`y=(m\lambda_2/2 D)/(d)`

The third-order bright fringe (m= 3) of wavelength A coincides with the fourth dark fringe (m=4) of the wavelength B. Hence you have that:

`((3)\lambda_1D)/(d)=((4)\lambda_2D)/(d)\\\\\lambda_2=(3)/(4)\lambda_1=(3)/(4)(564nm)=423nm`

hence, the wavelength B is 423nm