Complete Question
A student locates a double-slit assembly 1.40 m from a reflective screen. The slits are separated by 0.0572 mm.
(a) Suppose the student aims a beam of yellow light, with a wavelength of 589 nm, toward the slit assembly, and this makes an interference pattern on the screen. What distance (in cm) separates the zeroth-order and first-order bright fringes (a.k.a. maxima)?
(b)
Now suppose that blue light (with
λ = 415 nm)
is used instead. What distance (in cm) will now separate the second-order and fourth-order bright fringes?
Answer:
a
The distance of separation is ![z_1 - z_o = 1.44cm](https://tex.z-dn.net/?f=z_1%20-%20z_o%20%3D%201.44cm)
b
The distance of separation is ![z_4 - z_2 = 2.031cm](https://tex.z-dn.net/?f=z_4%20-%20z_2%20%20%3D%20%202.031cm)
Explanation:
From the question we are told that
The distance from the screen is ![D = 1.40m](https://tex.z-dn.net/?f=D%20%3D%201.40m)
The slit separation is ![d = 0.0572 mm = 0.0572 *10^{-3} m](https://tex.z-dn.net/?f=d%20%3D%200.0572%20mm%20%3D%200.0572%20%2A10%5E%7B-3%7D%20m)
The wavelength of the yellow light is ![\lambda_y = 598nm](https://tex.z-dn.net/?f=%5Clambda_y%20%3D%20598nm)
The distance of a fringe from the central maxima is mathematically represented as
![z_n = n \frac{\lambda_y D}{d}](https://tex.z-dn.net/?f=z_n%20%20%3D%20n%20%5Cfrac%7B%5Clambda_y%20D%7D%7Bd%7D)
Where n is the order of the fringe so the distance of separation between
The distance that separates first order from zeroth order bright fringe can be evaluated as
![z_1 - z_o = (1 - 0 ) \frac{\lambda_y D}{d}](https://tex.z-dn.net/?f=z_1%20-%20z_o%20%3D%20%281%20-%200%20%29%20%5Cfrac%7B%5Clambda_y%20D%7D%7Bd%7D)
Substituting values
![z_1 - z_o = (1 - 0 ) \frac{590*10^{-9} 1.40}{0.0572 *10^{-3}}](https://tex.z-dn.net/?f=z_1%20-%20z_o%20%3D%20%281%20-%200%20%29%20%5Cfrac%7B590%2A10%5E%7B-9%7D%201.40%7D%7B0.0572%20%2A10%5E%7B-3%7D%7D)
![z_1 - z_o = 0.0144m](https://tex.z-dn.net/?f=z_1%20-%20z_o%20%3D%200.0144m)
Converting to cm
![z_1 - z_o = 0.0144m = 0.0144*100 = 1.44cm](https://tex.z-dn.net/?f=z_1%20-%20z_o%20%3D%200.0144m%20%3D%20%200.0144%2A100%20%3D%201.44cm)
b
The wavelength of blue light is ![\lambda _b](https://tex.z-dn.net/?f=%5Clambda%20_b)
So the distance that separates second order from fourth order bright fringe can be evaluated as
![z_4 - z_2 = (4 - 2 ) \frac{\lambda_y D}{d}](https://tex.z-dn.net/?f=z_4%20-%20z_2%20%3D%20%284%20-%202%20%29%20%5Cfrac%7B%5Clambda_y%20D%7D%7Bd%7D)
Substituting values
![z_4 - z_2 = (4 - 2 ) \frac{415*10^{-9} 1.40}{0.0572 *10^{-3}}](https://tex.z-dn.net/?f=z_4%20-%20z_2%20%3D%20%284%20-%202%20%29%20%5Cfrac%7B415%2A10%5E%7B-9%7D%201.40%7D%7B0.0572%20%2A10%5E%7B-3%7D%7D)
![z_4 - z_2 = 0.02031 \ m](https://tex.z-dn.net/?f=z_4%20-%20z_2%20%3D%200.02031%20%5C%20m)
Converting to cm
![z_4 - z_2 = 0.02031m = 0.02031*100 = 2.031cm](https://tex.z-dn.net/?f=z_4%20-%20z_2%20%3D%200.02031m%20%3D%20%200.02031%2A100%20%3D%202.031cm)