A thin film of soap solution is illuminated by white light. In the reflected
light, two dark consecutive overlapping fringes are observed
corresponding to wavelengths 6.1 x 10^-7 m and 6.0 x 10^
-7 m. The
refractive index for soap solution is 4/3. Calculate the thickness of the
film.
To solve this problem, we'll use the formula for the thickness of a thin film:
t = (m + 1/2) * λ / (2 * n)
where:
t is the thickness of the film
m is the order number of the fringe
λ is the wavelength of light in the film
n is the refractive index of the film
First, let's calculate the difference in path lengths (∆d) between the two consecutive fringes that we observed.
∆d = (m + 1) * λ_1 - m * λ_2
where:
λ_1 = 6.1 x 10^-7 m is the wavelength of the first fringe
λ_2 = 6.0 x 10^-7 m is the wavelength of the second fringe
∆d = (1 + 1) * (6.1 x 10^-7) - 1 * (6.0 x 10^-7)
= 1.2 x 10^-7 m
Now let's calculate the average wavelength of light in the film, λ_avg:
λ_avg = (λ_1 + λ_2) / 2
= (6.1 x 10^-7 + 6.0 x 10^-7) / 2
= 6.05 x 10^-7 m
Finally, we can substitute these values into the formula to find the thickness of the film:
t = (∆d * λ_avg) / (2 * n)
= (1.2 x 10^-7 * 6.05 x 10^-7) / (2 * 4/3)
= (7.26 x 10^-14) / (8/3)
= (7.26 x 10^-14) * (3/8)
= 2.7375 x 10^-14 m
= 2.74 x 10^-14 m (rounded to two significant figures)
Therefore, the thickness of the film is 2.74 x 10^-14 m.
To calculate the thickness of the film, we can use the formula for the fringe width in a thin film given by:
Δx = λ / (2n)
Where:
Δx = fringe width
λ = wavelength of light in the medium
n = refractive index of the medium
Let's calculate the fringe width for both wavelengths:
For λ = 6.1 x 10^-7 m:
Δx1 = (6.1 x 10^-7 m) / (2 x 4/3)
Δx1 = (6.1 x 10^-7 m) / (8/3)
Δx1 = (6.1 x 10^-7 m) x (3/8)
Δx1 = 2.3 x 10^-7 m
For λ = 6.0 x 10^-7 m:
Δx2 = (6.0 x 10^-7 m) / (2 x 4/3)
Δx2 = (6.0 x 10^-7 m) / (8/3)
Δx2 = (6.0 x 10^-7 m) x (3/8)
Δx2 = 2.25 x 10^-7 m
Since the fringes are overlapping, the difference in fringe widths represents the thickness of the film. So, we can calculate the thickness as:
Thickness = Δx1 - Δx2
Thickness = (2.3 x 10^-7 m) - (2.25 x 10^-7 m)
Thickness = 0.05 x 10^-7 m
Thickness = 5 x 10^-9 m
Therefore, the thickness of the film is 5 x 10^-9 meters.
To calculate the thickness of the film, we can use the formula:
2t = (m + 1/2) * λ * n
where:
- t is the thickness of the film,
- m is the order of the fringe,
- λ is the wavelength of light, and
- n is the refractive index of the medium.
In this case, we are given two consecutive overlapping fringes with wavelengths of 6.1 x 10^-7 m and 6.0 x 10^-7 m, respectively.
First, let's find the order of the fringes. We know that the difference in path lengths between the two interfering rays results in destructive interference. For constructive interference, the path difference must be an integer multiple of the wavelength. Since the fringes are dark, we can conclude that the path difference is half a wavelength. This means that the order of the fringes (m) is an even number.
Now, let's calculate the thickness of the film using the formula above for one of the fringes.
For the first fringe with a wavelength of 6.1 x 10^-7 m:
2t = (m + 1/2) * λ * n
m = 2
λ = 6.1 x 10^-7 m
n = 4/3
Substituting the values into the formula:
2t = (2 + 1/2) * (6.1 x 10^-7 m) * (4/3)
2t = (5/2) * (6.1 x 10^-7 m) * (4/3)
2t ≈ 20 x 10^-7 m
Simplifying the equation:
2t ≈ 20 x 10^-7 m
t ≈ 10 x 10^-7 m
t ≈ 10^-6 m
Therefore, the thickness of the film is approximately 10^-6 m, or 1 micrometer.