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.