Relating Real and Apparent Depth to Refractive Index

Hello! Welcome to your sixth lesson in the Ray Optics module.

In our previous lesson, we established Snell's Law as the fundamental rule governing refraction. You learned how to derive it from Fermat's Principle of Least Time and apply it to calculate the path of light across different media.

Today, we will use Snell's Law to explain a very common optical illusion: why an object submerged in water appears to be at a shallower depth than it actually is. By the end of this lesson, you will be able to derive the relationship between real depth, apparent depth, and refractive index. This derivation is a classic application of the principles we've just covered and a perfect example of how geometry and physics come together.

1. Defining Real and Apparent Depth

Before we dive into the mathematics, let's clarify our terms.

Definition of Real Depth and Apparent Depth

This brief reading from BYJU'S clearly defines the concepts of real and apparent depth.

Please read the two short sections titled 'Real Depth' and 'Apparent Depth'. Focus on understanding the distinction between the actual position of the object and its perceived virtual position.

As the text explains:

Real Depth ( $d_{r e a l}$ ): The actual distance of an object from the surface of the medium.
Apparent Depth ( $d_{a pp}$ ): The depth at which the virtual image of the object appears to be located when viewed from a different medium.

The reason for this discrepancy is the refraction of light at the interface between the two media. Your brain assumes that light travels in straight lines. When light rays from the submerged object bend away from the normal as they exit the water, your brain traces these bent rays back in a straight line to a point above the actual object, creating a virtual image.

Derivation of Apparent Depth Formula — This diagram illustrates how light rays originating from an object (the fish) in a denser medium (water, $n_1$) refract at the surface and enter a rarer medium (air, $n_2$). An observer perceives the object at the location of the virtual image, which is at a shallower 'apparent depth' ($d_{app}$) than the 'real depth' ($d_{real}$).

2. Deriving the Relationship

Now, let's formalize this relationship with a rigorous derivation. Our goal is to connect $d_{r e a l}$ , $d_{a pp}$ , and the refractive indices of the two media involved. The key steps will be applying Snell's law and using a crucial geometric approximation for near-normal viewing.

The following video provides an excellent visual walkthrough of the derivation.

Refraction : case-1 derivation of Real & Apparent Depth expression | Grade 12 physics

This video by Make Me Scientific will guide you through the geometric setup and the step-by-step application of Snell's Law to derive the final expression.

Watch the video from the beginning until 08:18. Pay close attention to: How the ray diagram is constructed, and how the angles of incidence (i) and refraction (r) are identified using geometry (alternate interior angles). The introduction of the small angle approximation (paraxial approximation). The application of Snell's Law. How tan(i) and tan(r) are expressed in terms of real and apparent depths, leading to the final formula.

Let's consolidate that derivation. Consider the diagram above, with an object in a medium of refractive index $n_{1}$ (e.g., water) and an observer in a medium of refractive index $n_{2}$ (e.g., air).

Snell's Law: At the point of refraction on the surface, Snell's law applies:
$n_{1} sin i = n_{2} sin r$
where $i$ is the angle of incidence and $r$ is the angle of refraction.
The Small Angle Approximation: For an observer looking down almost vertically, the rays entering the eye are very close to the normal. These are called paraxial rays. For such rays, the angles $i$ and $r$ are very small. In calculus, you know that for a small angle $θ$ (measured in radians), the Taylor series expansion gives $sin θ \approx θ$ and $tan θ \approx θ$ . Therefore, for small angles, we can make the powerful approximation:
$sin θ \approx tan θ$
Applying this to Snell's Law gives:
$n_{1} tan i \approx n_{2} tan r$
Geometry: From the geometry of the triangles formed by the rays:
- In the large triangle with angle $i$ , $tan i = \frac{opposite}{adjacent} = \frac{x}{d _{r e a l}}$ .
- In the smaller triangle with angle $r$ , $tan r = \frac{opposite}{adjacent} = \frac{x}{d _{a pp}}$ .
Substitution and Final Formula: Substituting these geometric expressions back into our approximated Snell's Law:
$n_{1} (\frac{x}{d _{r e a l}}) = n_{2} (\frac{x}{d _{a pp}})$
The horizontal distance $x$ cancels out, leaving us with the general relationship:
$\frac{n _{1}}{d _{r e a l}} = \frac{n _{2}}{d _{a pp}}$
Or, rearranging for apparent depth:
$d_{a pp} = d_{r e a l} \cdot (\frac{n _{2}}{n _{1}})$
Where $n_{1}$ is the refractive index of the object's medium and $n_{2}$ is the refractive index of the observer's medium.

3. Analyzing the Result

The formula $d_{a pp} = d_{r e a l} \cdot (n_{2} / n_{1})$ is very instructive.

Case 1: Object in Denser Medium (e.g., fish in water, observer in air)
Here, $n_{1} > n_{2}$ (e.g., $n_{w a t er} \approx 1.33$ , $n_{ai r} \approx 1.0$ ).
This means the fraction $(n_{2} / n_{1}) < 1$ , so $d_{a pp} < d_{r e a l}$ . The object appears shallower, which matches our everyday experience.
In the common case where the observer is in air ( $n_{2} = 1$ ) and the object is in a medium of refractive index $n$ ( $n_{1} = n$ ), the formula simplifies to:
$d_{a pp} = \frac{d _{r e a l}}{n} or n = \frac{d _{r e a l}}{d _{a pp}}$
This is the version of the formula you will most often use in problems.
Case 2: Object in Rarer Medium (e.g., bird in air, observer underwater)
Here, $n_{1} < n_{2}$ (e.g., $n_{ai r} < n_{w a t er}$ ).
This means the fraction $(n_{2} / n_{1}) > 1$ , so $d_{a pp} > d_{r e a l}$ . The bird would appear farther away to the fish than it actually is.

Apparent Shift
The vertical distance between the real and apparent position of the object is called the apparent shift, $s$ .
$s = d_{r e a l} - d_{a pp}$
Substituting our formula for $d_{a pp}$ :
$s = d_{r e a l} - d_{r e a l} (\frac{n _{2}}{n _{1}})$
$s = d_{r e a l} (1 - \frac{n _{2}}{n _{1}})$
For the common case of viewing an object in a medium of index $n$ from air, this becomes $s = d_{r e a l} (1 - 1/ n)$ .

Test your understanding!

A fish is looking up at a dragonfly hovering 1 meter above the surface of the water ( $n_{w a t er} = 4/3$ ). From the fish's perspective, how high does the dragonfly appear to be?

Show answer

This is an example of Case 2, where the object is in the rarer medium and the observer is in the denser medium.

Real depth (in this case, real height): $d_{r e a l} = 1.0$ m.
Refractive index of object's medium (air): $n_{1} = 1.0$ .
Refractive index of observer's medium (water): $n_{2} = 4/3$ .

Using the general formula:
$d_{a pp} = d_{r e a l} \cdot (\frac{n _{2}}{n _{1}})$
$d_{a pp} = 1.0 m \cdot (\frac{4/3}{1.0})$
$d_{a pp} = \frac{4}{3} m \approx 1.33 m$

From the fish's perspective, the dragonfly appears to be 1.33 meters above the water surface, which is higher than its actual position.

Conclusion

In this lesson, we have derived and analyzed one of the direct consequences of Snell's Law.

Key Takeaways:
- Refraction at a plane surface causes a submerged object's apparent depth to be different from its real depth.
- Using Snell's Law combined with the small angle approximation ( $sin θ \approx tan θ$ ), we derived the general relationship: $d_{a pp} = d_{r e a l} \cdot (n_{o b ser v er} / n_{o bj ec t})$ .
- For an object in a denser medium viewed from a rarer medium, it appears shallower ( $d_{a pp} < d_{r e a l}$ ).
- For an object in a rarer medium viewed from a denser medium, it appears farther away ( $d_{a pp} > d_{r e a l}$ ).

Preview of the Next Lesson:
We've seen that when light goes from a denser to a rarer medium, it bends away from the normal. An interesting question arises: what happens if we keep increasing the angle of incidence? Is there a point where the light bends so much that it doesn't escape the first medium at all? The answer is yes, and this leads to the phenomenon of Total Internal Reflection. In the next lesson, you will derive the condition for total internal reflection and the expression for the critical angle.

Recommended resources

1. Defining Real and Apparent Depth

2. Deriving the Relationship

3. Analyzing the Result

Conclusion