Coherent light with wavelength 608 nm passes through two very narrow slits, and the interference pattern is observed on a screen a distance of 3.00 m from the slits. The first-order bright fringe is a distance of 4.84 mm from the center of the central bright fringe.
For what wavelength of light will thefirst-order dark fringe be observed at this same point on thescreen?
Express your answer in micrometers(not in nanometers).

Explanation:

In the double-slit interference, light passes through a double slit and produce a pattern of alternating bright and dark fringes on a distant screen. This pattern is due to the combined effect of the diffraction of each slit + the interference of the light coming from the two slits.

The condition to observe a maximum (bright fringe), so costructive interference, in the distant screen, is:

where:

y is the distance of the m-th maximum from the central fringe

is the wavelength of the light used

D is the distance of the screen from the slits

d is the separation between the slits

In this problem, we know that:

is the wavelength of light used

is the distance of the screen

is the distance of the first maximum (first-order bright fringe) from the central pattern, so when

m = 1

Solving for d, we find the separation of the slits:

The first dark fringe on the screen instead is given by the formula

where

is the wavelength of the new light

Here we want the first dark fringe of the new light to be coincident to the first bright fringe of the previous light, so

Therefore, solving for ,

you'd use the equation that establishes a relationship between  energy  and  wavelength, which looks like this

e=h⋅cλ, where

λ  - the wavelength of the photon; c  - the speed of light, usually given as  3⋅108m/s; h  -  planck's constant, equal to  6.626⋅10−34j s

plug the value you were given for the energy of the photon in the above equation to get the wavelength

e=h⋅cλ⇒λ=h⋅ce

λ=6.626⋅10−34js⋅3⋅108ms−1.257⋅10−24j=1.581⋅10−1m