A swimming pool filled with water to its edge, the depth of the water is 2m and an electric lamp is placed at a distance equals 8m from the edge of the pool over a column its height 6m. Calculate the length of non-apparent part from the pool bottom with the light does not reach to it, given the refractive index of water is 4?3.
I believe that the refractive index of water is n=1.3
Incidence angle α.
sin α =8/sqrt(8²+6)=0.8
sinα/sinβ=n =>
sinβ= sinα/n =0.8/1.3=0.61 =>β= 38°
x=h•tan β=2•tan38°=1.56 m
To calculate the length of the non-apparent part from the pool bottom, we need to consider the refraction of light as it passes from air to water and enters the pool.
The first step is to calculate the apparent height of the lamp when viewed from within the water. This can be done using the formula for apparent depth:
Apparent depth = Actual depth / Refractive index
In this case, the actual depth of the water is given as 2m, and the refractive index of water is 4/3. So the apparent depth of the water is:
Apparent depth = 2m / (4/3) = 1.5m
Next, we need to calculate the angle of incidence of light at the water surface. This can be done using trigonometry. The angle of incidence (θ1) is given by:
sin(θ1) = height of the column / distance from the edge of the pool to the column
In this case, the height of the column is given as 6m, and the distance from the edge of the pool to the column is given as 8m. So we have:
sin(θ1) = 6m / 8m = 0.75
To find the angle, we can take the inverse sine (sin^-1) of 0.75:
θ1 = sin^-1(0.75) = 48.59°
Now, we can use Snell's law to calculate the angle of refraction (θ2) as the light enters the water:
n1 * sin(θ1) = n2 * sin(θ2)
In this case, the refractive index of air is generally taken as 1, and the refractive index of water is 4/3. So we have:
1 * sin(48.59°) = (4/3) * sin(θ2)
Solving for sin(θ2), we get:
sin(θ2) = (1 * sin(48.59°)) / (4/3) = 0.4855
To find the angle, we can take the inverse sine (sin^-1) of 0.4855:
θ2 = sin^-1(0.4855) = 29.47°
Finally, we can calculate the length of the non-apparent part from the pool bottom by subtracting the apparent depth of the water from the total depth of the pool:
Length of non-apparent part = Actual depth - Apparent depth
= 2m - 1.5m
= 0.5m
Therefore, the length of the non-apparent part from the pool bottom where the light does not reach is 0.5 meters.