An aquarium filled with water has flat glass sides whose index of refraction is 1.48. A beam of light from outside the aquarium strikes the glass at a 43.5º angle to the perpendicular . The angle this light ray enters the glass is 27.7º, what is the angle of this light ray when it enters the water?
The angle in the glass ( r ) related to the angle of incidence (i) as
sin i/sin r = n2/n1
sin r = n1•sin i/n2 = 1•sin 43.5/1.48 = =0.465
r = arcsin 0.465= 27.7degr.
Because the surfaces are parallel, the refraction angle from the first surface is the incident angle at the second surface.
sin r/sin φ =n3/n2
sin φ = n2•sin r/n3 =1.48•0.465/1.33 =0.517,
φ = arcsin 0.517 =31.16 degr.
To solve this problem, we can use Snell's law, which relates the angles of incidence and refraction to the indices of refraction of the two mediums involved.
Snell's law is given by:
n₁ * sin(θ₁) = n₂ * sin(θ₂)
Where:
n₁ = index of refraction of the first medium (in this case, air)
n₂ = index of refraction of the second medium (in this case, glass)
θ₁ = angle of incidence
θ₂ = angle of refraction
Let's break down the problem step by step:
1. We are given that the angle of incidence is 43.5º relative to the perpendicular. To find the angle of incidence relative to the normal (which is the line perpendicular to the surface of the glass), we subtract the angle of incidence from 90º:
θ₁ = 90º - 43.5º = 46.5º
2. We also know that the index of refraction of the glass is 1.48.
3. We want to find the angle of refraction when the light ray enters the water. Since we are going from glass to water, we'll use the following formula:
n₁ * sin(θ₁) = n₂ * sin(θ₂)
4. The index of refraction of air is approximately 1. So, plugging in the values we know:
1 * sin(46.5º) = 1.48 * sin(θ₂)
5. Rearranging the equation and solving for θ₂:
sin(θ₂) = (1 * sin(46.5º)) / 1.48
θ₂ = sin^(-1)((1 * sin(46.5º)) / 1.48)
Calculating θ₂:
θ₂ ≈ sin^(-1)((1 * 0.7193) / 1.48)
θ₂ ≈ sin^(-1)(0.4865)
θ₂ ≈ 29º
Therefore, the angle of the light ray when it enters the water is approximately 29º.