Interference effects are produced at point P on a screen as a result of direct rays from a 518 nm source and reflected rays off the mirror.If the source is 99 m to the left of the screen, and 1.15 cm above the mirror, calculate the distance y (in millimeters) to the first dark band above the mirror
To calculate the distance to the first dark band above the mirror, we need to consider the path difference between the direct and reflected rays.
The path difference (ΔP) is given by:
ΔP = 2d * cos(θ)
where d is the distance between the source and the mirror, and θ is the angle of incidence.
First, let's calculate the angle of incidence (θ). Given that the source is 99 meters to the left of the screen and the mirror is 1.15 cm above the mirror, we can use trigonometry to find θ.
θ = arctan(h / d)
where h is the height of the mirror above the screen, and d is the distance between the source and the screen.
Plugging in the values:
θ = arctan(1.15 cm / 99 m)
Now that we have the angle of incidence (θ), we can calculate the path difference (ΔP).
ΔP = 2 * d * cos(θ)
Substituting the given values:
ΔP = 2 * 99 m * cos(arctan(1.15 cm / 99 m))
Now we can calculate the distance (y) to the first dark band above the mirror. The dark band occurs when ΔP equals half the wavelength (λ/2).
y = (ΔP * λ) / (2 * π)
Substituting the given wavelength (λ = 518 nm) and the calculated path difference (ΔP), we can solve for y.
y = ((2 * 99 m * cos(arctan(1.15 cm / 99 m))) * 518 nm) / (2 * π)
Converting the result to millimeters, we have the distance (y) to the first dark band above the mirror.
To calculate the distance to the first dark band above the mirror, we need to use the concept of interference.
1. Calculate the path difference:
The path difference between the direct rays and reflected rays can be calculated using the formula:
Path Difference = 2 * d * sin(θ)
where d is the distance between the source and the mirror, and θ is the angle of incidence.
We know that the distance between the source and the screen is 99 m, so the distance between the source and the mirror would be half of that: d = 99 m / 2 = 49.5 m.
Also, we know that the source is located 1.15 cm above the mirror. To calculate the angle of incidence, we can use trigonometry:
tan(θ) = opposite / adjacent
tan(θ) = 1.15 cm / 49.5 m
θ = arctan(1.15 cm / 49.5 m)
2. Calculate the wavelength in meters:
The wavelength is given as 518 nm. To convert it to meters, we divide by 10^9:
λ = 518 nm / (10^9) m = 5.18 × 10^(-7) m
3. Calculate the distance to the first dark band:
To find the distance to the first dark band, we need to use the condition for destructive interference:
Path Difference = (m + 1/2) * λ
where m is an integer representing the order of the dark band.
Substitute the values into the equation and solve for y:
2 * d * sin(θ) = (m + 1/2) * λ
y = (2 * d * sin(θ) - (1/2) * λ) / λ
Insert the known values:
y = (2 * 49.5 m * sin(θ) - (1/2) * 5.18 × 10^(-7) m) / 5.18 × 10^(-7) m
Calculate the value of sin(θ) using the previously determined value of θ.
4. Convert the distance y to millimeters:
Multiply the value of y by 1000 to convert it from meters to millimeters.
And that's it! You have calculated the distance to the first dark band above the mirror (y) in millimeters.