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.

What is Apparent Depth?

Apparent depth is a fake depth, but it looks real depth of any object when the person looking at an object is in a different medium than the object. It is an effect because of which we get to see many types of illusion between two transparent mediums when the observer and target(object) both are in different mediums.

Definition of Apparent Depth

Apparent depth is a term used in optics to describe how an object submerged in a transparent medium appears to be located at a different depth than its actual position. It’s an optical illusion that occurs due to the bending of light rays as they pass from one medium to another with different refractive indices.

Apparent Depth in Different Medium

As we know, the phenomenon of apparent depth occurs due to the bending of light (refraction) while travelling from one medium to another. When dealing with mediums, we define two types of them: denser medium and rarer medium, which are relative terms. A medium with a higher refractive index is denser compared to a medium with a lower refractive index, while the same medium can be considered rarer when compared to another medium with a higher refractive index.

When light rays pass from a denser to a rarer medium, they get refracted away from the normal (the common normal to both mediums), causing the light rays to appear to be coming from a virtual object formed above the actual object. If the light rays pass from a rarer to a denser medium, they move towards the normal, causing the light rays to appear to be coming from a virtual object formed below the actual object. This is how apparent depth changes according to denser and rarer mediums.

Laws of Refraction and Apparent Depth

Apparent depth can be explained using the laws of refraction, specifically Snell’s Law. According to Snell’s Law, the angle of incidence and the angle of refraction are related to the refractive indices of the two media:

μ₁ . sin(θ₂) = μ₂ . sin(θ₁)

Where,

• μ₁ and μ₂ are the refractive indices of the two media,
• θ₁ is the angle of incidence in the first medium, and
• θ₂ is the angle of incidence in the first medium.

Refere illustration for further understanding.

Apparent Depth Formula

D_(Apparent) = D_(Real) / μ₂₁ 

Where,

• D_(apparent) is the apparent depth of an object,
• D_(real) is the real depth of an object, and
• μ₂₁ is the relative refractive index of medium 2 with respect to medium 1 and is given as:

μ₂₁ = μ₂ / μ₁

Derivation of Apparent Depth Formula

To derive the formula for apparent depth we will follow the figure given below:

A light ray originating from the coin gets refracted and moves away from the normal. When it is traced back, we can determine the position of the virtual object. To determine the position of the virtual object, at least two rays of light are needed. In this diagram, the primary purpose is to illustrate the concept of apparent depth.

We can perform below calculations referring the figure.

cos(90−θ₁) = sin(θ₁) = X / D_(apparent) —–(1)

and cos(90−θ₂) = sin(θ₂) = X / D_(real) —–(2)

Dividing (1) and (2) ,

sin(θ₁) / sin(θ₂) = D_(apparent) / D_(real)

Using Snell’s Law

μ₁₂ = 1/μ₂₁  = μ₁/μ₂  = sin(θ₁)/sin(θ₂) 

⇒ D_(apparent) / D_(real) = 1/μ₂₁

⇒ D_{(apparent) =} D_(real) /μ₂₁

Which is the required formula.

Apparent and Real Depth

Apparent and Real Depth are the depths of any object when submerged under any different medium. The actual depth of a submerged object is called real depth, and the perceived depth of any object is due to the change of medium, which is called Apparent Depth. In addition to this, there are some more key differences between Apparent and real depth. These differences are listed in the following table:

Aspect	Real Depth	Apparent Depth
Definition	The actual physical distance from an observer to an object submerged in a transparent medium.	The perceived distance from an observer to an object submerged in a transparent medium, as it appears to the observer.
Calculation	Determined based on the actual physical dimensions of the object and its position in the medium.	Influenced by the refraction of light as it passes through the boundary between two media with different optical densities.
Value	Constant and does not change with observation or medium.	Variable and can change depending on the angle of observation, the refractive index of the medium, and the position of the object.
Examples	If an object is 10 cm below the water’s surface, its real depth is 10 cm.	If an object is 10 cm below the water’s surface, its apparent depth may be less than 10 cm when viewed from above due to refraction.
Use in Optics	Used in calculations involving Snell’s law, lens equations, and other optical phenomena.	Used to explain optical illusions like the bending of a straw in a glass of water or the appearance of objects “closer” in water.
Units	Measured in meters (m), centimetres (cm), or any appropriate length unit.	Measured in the same units as real depth (m, cm), but may appear different due to optical effects.

Factors Affecting Apparent Depth

Several factors influence apparent depth:

• Change in Medium: Apparent depth changes when an object moves from one medium to another with a different refractive index.
• Angle of Incidence: The angle at which light enters the interface between two media affects the degree of bending and, consequently, the apparent depth.
• Speed of Light in a Medium: Basically refractive index is affected by by the speed of light in a medium and hence apparent depth also gets impacted.
• Wavelength of Light: The apparent depth also depends on the wavelength of light. Different colours of light bend by different amounts, which can create colourful optical effects.

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.

Practice Problems on Apparent Depth

Problem 1: A coin lies at the bottom of a swimming pool with a depth of 2 meters. Calculate the apparent depth of the coin when viewed from above the water. [refractive index of water is 1.33]

Problem 2: A scuba diver is exploring a lake with a refractive index of 1.4. If the diver is actually 20 meters deep in the lake, calculate the apparent depth of the diver when seen from the surface.

Apparent Depth – FAQs

1. Define Apparent Depth.

Apparent depth is a concept in physics related to the perceived depth of an object submerged in a transparent medium, such as water or glass.

2. How is Apparent Depth Measured?

Apparent depth can be  measured using the relation i.e., D_Apparent = D_Real / μ₂₁

Where,

• D_Apparent  is the apparent depth of an object,
• D_Real is the real depth of an object, and
•  μ₂₁ is the relative refractive index. 

3. What are Real and Apparent Depths?

Real depth is the actual physical distance between objects. Apparent depth is the perceived depth, affected by refraction when light passes through different mediums, creating optical illusions.

4. What is the Formula of Actual Depth?

There isn’t a specific formula for actual depth because it’s the physical, real-world depth of an object. But with the formula of apparent depth we can find real depth as well if apparent depth is given i.e., D(apparent) = D(real) / μ₂₁

5. What is the Relation between Apparent Depth and Refractive Index?

Apparent depth is inversely proportional to the refractive index of the medium. As the refractive index increases, the apparent depth decreases, and vice versa. This relationship is described by Snell’s Law.

6. How do you find Apparent Depth in Physics?

Apparent Depth can be found using the formula i.e., D_Apparent = D_Real / μ₂₁