In Young's double-slit experiment, the position of bright and dark fringes depends on the wavelength of light and the distance between the slits and the screen. When the experiment is performed in a medium other than air, the refractive index of the medium affects the wavelength of light. The relationship between the wavelengths in air (λ_air) and in the medium (λ_medium) is given by:
λ_medium = λ_air / n
Where:
λ_medium is the wavelength of light in the medium.
λ_air is the wavelength of light in air.
n is the refractive index of the medium.
Now, you mentioned that the 8th bright fringe in the medium corresponds to the 5th dark fringe in air. In Young's double-slit experiment, the position of bright fringes is given by:
dsin(θ) = mλ
Where:
d is the distance between the slits.
θ is the angle at which the fringe is observed.
m is the order of the fringe (e.g., 1st, 2nd, 3rd, etc.).
λ is the wavelength of light.
For the 5th dark fringe in air, m = 5, and for the 8th bright fringe in the medium, m = 8.
Since the 5th dark fringe in air corresponds to the 8th bright fringe in the medium, we can write:
dsin(θ_air) = 5λ_air
dsin(θ_medium) = 8λ_medium
Now, we can use the relationship between the wavelengths in air and the medium to relate these two equations:
dsin(θ_medium) = 8(λ_air / n)
We can divide the second equation by the first one:
(8*(λ_air / n)) / (5*λ_air)
Now, we can cancel out λ_air:
(8 / 5n) = sin(θ_medium) / sin(θ_air)
We know that sin(θ_medium) and sin(θ_air) are both related to the same angle θ, so their ratio remains the same. Therefore:
(8 / 5n) = constant
To find n, we need to calculate this constant. From the problem statement, it's given that the 8th bright fringe in the medium corresponds to the 5th dark fringe in air, so:
8 / 5n = 1
Now, solve for n:
n = 8 / 5
n = 1.6
So, the refractive index of the medium is approximately 1.6.
None of the answer choices provided matches exactly, but the closest one to 1.6 is option C, which is 1.59. Therefore, option C (1.59) is the best approximation for the refractive index of the medium.
