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In a double-slit experiment, at a certain point on the screen, the path difference between the two interfering waves is 1/8th of a wavelength. The ratio of the intensity of light at that point to that at the centre of a bright fringe is
- a)0.568
- b)0.760
- c)0.853
- d)0.672
Correct answer is option 'C'. Can you explain this answer?
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In a double-slit experiment, at a certain point on the screen, the pat...
Given:
- In a double-slit experiment
- Path difference between the two interfering waves at a certain point on the screen is 1/8th of a wavelength
To find:
- The ratio of the intensity of light at that point to that at the center of a bright fringe
Explanation:
The intensity of light in a double-slit experiment can be calculated using the formula:
I = 4 * I0 * cos^2(π * d * sinθ / λ)
Where:
- I is the intensity of light at a certain point on the screen
- I0 is the intensity of light at the center of a bright fringe
- d is the distance between the two slits
- θ is the angle between the line joining the point on the screen and the center of the pattern and the line joining the two slits
- λ is the wavelength of light
Step 1:
Calculate the path difference between the two interfering waves at the given point on the screen.
Path difference = 1/8th of a wavelength = λ/8
Step 2:
Using the path difference, calculate the phase difference between the two interfering waves.
Phase difference = 2π * path difference / wavelength = 2π * (λ/8) / λ = π/4
Step 3:
Calculate the angle θ using the formula:
sinθ = phase difference / distance between the point on the screen and the center of the pattern
Since sinθ is small for small angles, we can approximate sinθ ≈ θ in radians.
θ = phase difference / distance between the point on the screen and the center of the pattern
Step 4:
Substitute the calculated value of θ in the intensity formula to find the ratio of the intensity at the given point to the intensity at the center of a bright fringe.
I / I0 = 4 * cos^2(π * d * sinθ / λ) / 4 * cos^2(0) = cos^2(π * d * θ / λ) / cos^2(0) = cos^2(π * d * θ / λ)
Step 5:
Calculate the value of cos^2(π * d * θ / λ) using the given values and substitute in the formula.
I / I0 = cos^2(π * d * θ / λ) = cos^2(π * d * (π/4) / λ) = cos^2(π^2 * d / (4 * λ))
Step 6:
Since the value of d/λ is not given, we can assume d = λ/2 to simplify the calculation.
I / I0 = cos^2(π^2 * (λ/2) / (4 * λ)) = cos^2(π/8)
Step 7:
Using the trigonometric identity cos^2(x) = (1 + cos(2x)) / 2, we can simplify further.
I / I0 = (1 + cos(2 * (π/8))) / 2 = (1 + cos(π/4)) / 2
Step 8:
Using
- In a double-slit experiment
- Path difference between the two interfering waves at a certain point on the screen is 1/8th of a wavelength
To find:
- The ratio of the intensity of light at that point to that at the center of a bright fringe
Explanation:
The intensity of light in a double-slit experiment can be calculated using the formula:
I = 4 * I0 * cos^2(π * d * sinθ / λ)
Where:
- I is the intensity of light at a certain point on the screen
- I0 is the intensity of light at the center of a bright fringe
- d is the distance between the two slits
- θ is the angle between the line joining the point on the screen and the center of the pattern and the line joining the two slits
- λ is the wavelength of light
Step 1:
Calculate the path difference between the two interfering waves at the given point on the screen.
Path difference = 1/8th of a wavelength = λ/8
Step 2:
Using the path difference, calculate the phase difference between the two interfering waves.
Phase difference = 2π * path difference / wavelength = 2π * (λ/8) / λ = π/4
Step 3:
Calculate the angle θ using the formula:
sinθ = phase difference / distance between the point on the screen and the center of the pattern
Since sinθ is small for small angles, we can approximate sinθ ≈ θ in radians.
θ = phase difference / distance between the point on the screen and the center of the pattern
Step 4:
Substitute the calculated value of θ in the intensity formula to find the ratio of the intensity at the given point to the intensity at the center of a bright fringe.
I / I0 = 4 * cos^2(π * d * sinθ / λ) / 4 * cos^2(0) = cos^2(π * d * θ / λ) / cos^2(0) = cos^2(π * d * θ / λ)
Step 5:
Calculate the value of cos^2(π * d * θ / λ) using the given values and substitute in the formula.
I / I0 = cos^2(π * d * θ / λ) = cos^2(π * d * (π/4) / λ) = cos^2(π^2 * d / (4 * λ))
Step 6:
Since the value of d/λ is not given, we can assume d = λ/2 to simplify the calculation.
I / I0 = cos^2(π^2 * (λ/2) / (4 * λ)) = cos^2(π/8)
Step 7:
Using the trigonometric identity cos^2(x) = (1 + cos(2x)) / 2, we can simplify further.
I / I0 = (1 + cos(2 * (π/8))) / 2 = (1 + cos(π/4)) / 2
Step 8:
Using
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In a double-slit experiment, at a certain point on the screen, the path difference between the two interfering waves is 1/8th of a wavelength. The ratio of the intensity of light at that point to that at the centre of a bright fringe isa)0.568b)0.760c)0.853d)0.672Correct answer is option 'C'. Can you explain this answer?
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