In an experiment conducted using the Young's double-slit apparatus, the separation between the slits is 20 µm. A first-order constructive interference fringe appears at an angle of 2.5o from the zeroth order (central) fringe.
A. What wavelength of light is used in the experiment?
B. At what angle would the fourth-order (m = 4) bright fringe appear?
C. At what angle would the fourth-order (m=4) dark fringe appear?
y(max1) =mλL/d
λ=y(max1)•d/m•L
tanα=y(max1)/L
λ=tanα•d/m=tan2.5°•20•10⁻⁶/1=0.873•10⁻⁶ m
tanα=L •m/d=….
Beeef
To answer these questions, we can use the equation for the fringe spacing in Young's double-slit experiment:
λ = (m * d) / (s * L)
Where:
λ is the wavelength of light
m is the order of the fringe
d is the separation between the slits
s is the fringe spacing
L is the distance between the double-slit and the screen
Let's solve these questions step by step:
A. What wavelength of light is used in the experiment?
Given:
d = 20 µm (separation between the slits)
m = 1 (first-order fringe)
θ = 2.5° (angle from the zeroth order fringe)
We can first calculate the fringe spacing using the equation:
s = d * tan(θ)
s = 20 µm * tan(2.5°)
Now, we can rearrange the equation and solve for λ:
λ = (m * d) / (s * L)
Since the first-order fringe appears at an angle of 2.5°, m = 1. The distance L is not provided, so the exact wavelength of light cannot be calculated without it. However, if you know the value of L, you can substitute it into the equation along with the calculated values of d and s to find the wavelength of light used in the experiment.
B. At what angle would the fourth-order (m = 4) bright fringe appear?
Using the same equation, we can calculate the angle for the fourth-order fringe.
Given:
m = 4 (order of the fringe)
d = 20 µm (separation between the slits)
Knowing the value of m, we can find the fringe spacing by:
s = (m * λ * L) / d
Assuming we know the wavelength of light and the L value, we can substitute them into the equation to find the fringe spacing and then calculate the angle θ using the formula:
θ = atan(s / L)
C. At what angle would the fourth-order (m = 4) dark fringe appear?
The position of dark fringes can be determined using the formula:
sin(θ) = (m + 0.5) * λ / d
Given:
m = 4 (order of the fringe)
d = 20 µm (separation between the slits)
Similar to the previous question, if we know the wavelength of light used, we can rearrange the formula to solve for the angle θ:
θ = arcsin((m + 0.5) * λ / d)
Again, we need the exact value of λ and the L distance to calculate the angle accurately.