A double-slit interference pattern is created by two narrow slits spaced 0.21 mm apart. The distance between the first and the fifth minimum on a screen 55 cm behind the slits is 6 mm. What is the wavelength of the light used in this experiment?
Having trouble setting this up.
Thanks
To solve this problem, let's break it down step by step.
Step 1: Understand the problem
We have a double-slit interference pattern created by two narrow slits that are 0.21 mm apart. The distance between the first and the fifth minimum on the screen located 55 cm behind the slits is given as 6 mm. We need to find the wavelength of the light used in this experiment.
Step 2: Identify the relevant formula
The formula we'll be using is the formula for the double-slit interference pattern, which relates the wavelength of light, the distance between the slits, the angle at which the interference pattern is observed, and the distance between the consecutive minimums (or maximums) on the screen.
The formula is:
λ = (d * sinθ) / m
Where:
λ is the wavelength of light
d is the distance between the slits
θ is the angle at which the interference pattern is observed
m is the order of the minimum (or maximum)
Step 3: Determine the variables and values
From the problem, we are given:
d = 0.21 mm
θ = not given (we'll solve for this in Step 4)
m = 5-1 = 4 (since we are looking at the distance between the first and fifth minimums)
Step 4: Find the angle at which the interference pattern is observed
To find the angle θ, we can use the following formula:
θ = tan^(-1) ((L * m) / (d * D))
Where:
L is the distance between the slits and the screen (55 cm = 0.55 m)
m is the order of the minimum (or maximum)
d is the distance between the slits
D is the distance between the screen and the central maximum
In this case, since we're given the distance between the first and fifth minimum (6 mm), the distance between the screen and the central maximum (D) can be found by:
D = (m * λ * L) / d
Substituting the given values:
D = (4 * λ * 0.55) / 0.21
Step 5: Substitute the values into the formula
Now we can substitute the value of D into the formula for θ:
θ = tan^(-1) ((0.55 * 4) / 0.21) = tan^(-1) (10.476)
Step 6: Solve for the wavelength
Finally, using the formula for λ, we can solve for the wavelength:
λ = (d * sinθ) / m = (0.21 * sin(tan^(-1)(10.476))) / 4
Calculating this value will give us the wavelength of the light used in the experiment.