Solved Problem on Apparent Depth
Problem 1: A fish is swimming at a depth of 5 meters in a pond with a refractive index of 1.33. What is the apparent depth of the fish when viewed from above the water?
Solution:
Using the apparent depth formula:
D(apparent) = D(real) / μ21
⇒ D(apparent) = 5 / 1.33
⇒ D(apparent) = 3.76m
So, the fish appears to be at an apparent depth of approximately 3.76 meters.
Problem 2: A straw is placed vertically in a glass of water. If the straw appears to be bent at an angle of 30 degrees at the water’s surface, calculate the actual angle at which the straw is submerged in the water. [refractive index of water is 1.33]
Solution:
Refractive index of air (n1) is approximately 1 (for simplicity),
Refractive index of water (n2) is 1.33,
Angle of incidence in air (θ1) is 30°, and
We want to find the actual angle at which the straw is submerged in the water, which is θ2.
According to Snell’s Law:
n1 × sin(θ1) = n2 × sin(θ2),
⇒ sin(θ2) = (n1/n2) × sin(θ1).
⇒ sin(θ2) = (1 / 1.33) × sin(30°)
⇒ sin(θ2) = (0.7519) × 0.5, [As sin(30°) is 0.5]
⇒ sin(θ2) = 0.37595.
Now, to find θ2, take the inverse sine (sin-1) of 0.37595:
θ2 = sin-1(0.37595)
Use a calculator to find the sin-1, and you’ll get:
θ2 ≈ 22.07°
So, the actual angle at which the straw is submerged in the water is approximately 22.07 degrees.
Apparent Depth
Apparent Depth is an example of reflection of light from different mediums and can be observed in many areas. Such as when we drink water from a bottle or any vessel we have an illusion that the bottle’s base is very near to us, but after it gets empty we realize its actual depth. This illusion is a result of the phenomenon known as Apparent Depth. Apparent Depth plays a crucial role in our perception of objects submerged in liquids as there are many such examples like a fish in a pond appearing near to the surface, the bottom of the Swimming Pool appearing near to the surface, and many more.
In this article, we will dive deep into the phenomenon of apparent depth, exploring its definition, the laws of refraction governing it, its mathematical formulation, the factors affecting it, and solving a problem to solidify our understanding.
Table of Content
- What is Apparent Depth?
- Apparent Depth in Different Medium
- Apparent Depth Formula
- Apparent and Real Depth
- Factors Affecting Apparent Depth