Answer: [tex]135.29\º[/tex]
The figure attached will be helpful to understand the solution.
Here we see two cases, reflection and refraction of light.
According to the laws of reflection:
1. The incident ray, the reflected ray and the normal are in the same plane.
2. The angle of incidence is equal to the angle of reflection.
*Note the normal is perpendicular to the plane, with a [tex]90\º[/tex] angle with the surface
And this can be visualized in the figure, where the angle [tex]{\theta}_{1}[/tex] with which the incident ray hits the surface is equal to the angle with which this same ray is reflected.
On the other hand we have Refraction, a phenomenon in which the light bends or changes its direction when passing through a medium with a refractive index different from the other medium.
In this context, the Refractive index is a number that describes how fast light propagates through a medium or material.
According to Snell’s Law:
[tex]n_{1}sin(\theta_{1})=n_{2}sin(\theta_{2})[/tex] (1)
Where:
[tex]n_{1}=1[/tex] is the first medium refractive index (the air)
[tex]n_{2}=1.309[/tex] is the second medium refractive index (the ice)
[tex]\theta_{1}=25.5\º[/tex] is the angle of the incident ray
[tex]\theta_{2}[/tex] is the angle of the refracted ray
Now, firstly we have to find [tex]\theta_{2}[/tex] and then, by geometry, find [tex]\alpha[/tex] and [tex]\beta[/tex] which sum the angle between the reflected and the refracted light.
Finding [tex]\theta_{2}[/tex] from (1):
[tex](1)sin(25.5\º)=(1.309)sin(\theta_{2})[/tex]
[tex]sin(\theta_{2})=0.328[/tex]
[tex]\theta_{2}=arcsin(0.328)[/tex]
[tex]\theta_{2}=19.201\º[/tex] (2)
Remembering that the Normal makes a [tex]90\º[/tex] angle with the surface, we can say:
[tex]90\º=\theta_{1}+\beta[/tex] (3)
Finding [tex]\beta[/tex]:
[tex]\beta=90\º-25.5\º=64.5\º[/tex] (4)
Doing the same with [tex]\theta_{2}[/tex] and [tex]\alpha}[/tex]:
[tex]90\º=\theta_{2}+\alpha[/tex] (5)
Finding [tex]\alpha[/tex]:
[tex]\alpha=90\º-19.201\º=70.79\º[/tex] (6)
Adding both angles (4) and (6):
[tex]\alpha+\beta=70.79\º+64.5\º[/tex] (7)
[tex]\alpha+\beta=135.29\º[/tex]>>>This is the angle between the reflected and the refracted light.
