A cylindrical glass of radius r=3 cm is full of salt water of height h=10 cm with the water's refraction index changing linearly from the bottom n1=1.401 to the top n2=1.400.
A flashlight is positioned so that light shines horizontally onto the glass. If we look at the light beams that pass through the center axis of the glass and out the other side, we notice that the outgoing ray (after it leaves the glass on the other side) is bent by a small angle with respect to the horizontal axis. Find the magnitude of that angle in degrees.
Details and assumptions
You may neglect any refraction in the actual glass itself. All the refraction happens in the water.
To find the magnitude of the angle by which the outgoing ray is bent, we first need to calculate the angle of refraction at the top of the water column. The angle of refraction can be calculated using Snell's law:
n1 * sin(angle of incidence at bottom) = n2 * sin(angle of refraction at top)
In this case, the angle of incidence at the bottom is 0 degrees since the light is entering the water horizontally. Therefore, the equation simplifies to:
n1 * sin(0) = n2 * sin(angle of refraction at top)
Since sin(0) is 0, the equation further simplifies to:
0 = n2 * sin(angle of refraction at top)
This implies that the angle of refraction at the top of the water column is also 0 degrees.
Next, we calculate the angle of incidence at the bottom of the water column using the formula for the critical angle of refraction, which is the angle at which the light rays no longer pass from a higher refractive index to a lower refractive index:
angle of incidence at bottom = arcsin(n2/n1)
Plugging in the values for n1 and n2, we get:
angle of incidence at bottom = arcsin(1.400/1.401) ≈ 0.1435 degrees
Now, we can calculate the angle by which the outgoing ray is bent using the following formula:
angle of deviation = 2 * angle of incidence at bottom
Plugging in the value for the angle of incidence at the bottom, we get:
angle of deviation = 2 * 0.1435 ≈ 0.287 degrees
Therefore, the magnitude of the angle by which the outgoing ray is bent is approximately 0.287 degrees.