Refraction at Plane Surfaces: Laws, Apparent Depth, Critical Angle, Lateral Shift

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## What is Refraction?
When light travels from one medium (like air) into another (like water or glass), it changes speed and **bends**. This bending is called **refraction**. It happens because the speed of light depends on the medium.



### Key Terms:
- **Incident ray**: Light ray striking the boundary.
- **Refracted ray**: Light ray bending inside the second medium.
- **Normal**: An imaginary line perpendicular to the boundary at the point of incidence.
- **Angle of incidence (i)**: Angle between incident ray and normal.
- **Angle of refraction (r)**: Angle between refracted ray and normal.

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## Snell’s Law of Refraction
Snell’s law relates the angles of incidence and refraction to the speeds (or refractive indices) of the two media:

\[
n_1 \sin i = n_2 \sin r
\]

Where:
- \( n_1 \) = refractive index of medium 1 (e.g., air = 1.00)
- \( n_2 \) = refractive index of medium 2 (e.g., water = 1.33, glass = 1.50)
- \( i \) = angle of incidence
- \( r \) = angle of refraction

### Worked Example 1: Finding Angle of Refraction
*A light ray travels from air (n₁ = 1.00) into water (n₂ = 1.33) at an angle of incidence of 45°. Find the angle of refraction.*

**Solution:**
Using Snell’s law:
\[
1.00 \cdot \sin 45° = 1.33 \cdot \sin r
\]
\[
\sin r = \frac{\sin 45°}{1.33} = \frac{0.707}{1.33} \approx 0.532
\]
\[
r = \sin^{-1}(0.532) \approx 32.0°
\]

**Answer:** The angle of refraction is **32.0°**.

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## Apparent Depth vs. Real Depth
When you look at an object underwater, it appears **shallower** than it actually is. This is because light bends away from the normal when exiting water into air.

```figure
{"type":"height-distance","angle":30,"height":12,"distance":16,"object":"Coin","medium":"water","realDepth":{"label":"Real depth (12 cm)","color":"red"},"apparentDepth":{"label":"Apparent depth (~9 cm)","color":"blue"},"caption":"Coin appears shallower due to refraction (n_water = 1.33)"}

Formula:

$$\text{Apparent depth}=\frac{\text{Real depth}}{n}$$ Where $n$ = refractive index of the medium (e.g., water = 1.33).

Worked Example 2: Calculating Apparent Depth

A coin is placed at the bottom of a water tank (n = 1.33) at a real depth of 12 cm. How deep does it appear to an observer above the water?

Solution: $$\text{Apparent depth}=\frac{12}{1.33}\approx 9.02\text{ cm}$$

Answer: The coin appears 9.02 cm deep.

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Critical Angle and Total Internal Reflection

When light travels from a denser to a rarer medium (e.g., water to air), it bends away from the normal. If the angle of incidence is too large, the refracted ray disappears, and all light reflects back. This angle is called the critical angle (θ_c).

Formula:

$$\sin {\theta }_c=\frac{n_2}{n_1}$$

Worked Example 3: Finding Critical Angle

Find the critical angle for light traveling from glass (n = 1.50) to air (n = 1.00).

Solution: $$\sin {\theta }_c=\frac{1.00}{1.50}=0.667$$ $${\theta }_c={\sin }^{-1}(0.667)\approx 41.8°$$

Answer: The critical angle is 41.8°.

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Lateral Shift of Light

When light passes through a parallel-sided slab (e.g., glass slab), it emerges parallel to the incident ray but shifted sideways. This shift is called lateral displacement (d).

Formula:

$$d=t\cdot \frac{\sin (i-r)}{\cos r}$$ Where:

• $t$ = thickness of the slab
• $i$ = angle of incidence
• $r$ = angle of refraction

Worked Example 4: Calculating Lateral Shift

A light ray enters a glass slab (n = 1.50) of thickness 4 cm at an angle of 60°. Find the lateral shift if the angle of refraction is 34.1°.

Solution: $$d=4\cdot \frac{\sin (60°-34.1°)}{\cos 34.1°}=4\cdot \frac{\sin 25.9°}{\cos 34.1°}$$ $$d=4\cdot \frac{0.436}{0.828}\approx 2.11\text{ cm}$$

Answer: The lateral shift is 2.11 cm.

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Comparison Table: Refraction Scenarios

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Applications of Refraction

1. Optical Instruments: Lenses and prisms use refraction to focus light (e.g., microscopes, cameras).
2. Mirages: Light bending in hot air creates illusions of water.
3. Fiber Optics: Total internal reflection transmits data as light pulses.
4. Corrective Lenses: Glasses use refraction to correct vision (e.g., convex for farsightedness).
5. Rainbow Formation: Refraction and dispersion in raindrops split light into colors.

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Common Mistakes to Avoid

1. Assuming refraction only bends light toward the normal: It bends away when going from denser to rarer media.
2. Ignoring the critical angle: Total internal reflection only occurs when $\theta >{\theta }_c$.
3. Mixing up apparent and real depth: Apparent depth is smaller than real depth in water.
4. Forgetting Snell’s law units: Angles must be in degrees for sine calculations.

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Exam Tip: How to Score Full Marks

1. Always draw diagrams: Label angles (i, r), media (n₁, n₂), and directions clearly.
2. Use Snell’s law correctly: Write the formula and substitute values step-by-step.
3. Memorize critical angle formula: $\sin {\theta }_c=\frac{n_2}{n_1}$.
4. Practice numericals: Apparent depth and lateral shift questions are common.
5. Explain real-world examples: Relate refraction to daily life (e.g., swimming pools appearing shallower).

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NEB Board-Style Questions

Short Answer (5 marks)

1. A light ray travels from diamond (n = 2.42) to water (n = 1.33) at an angle of 30°. Calculate the angle of refraction. Solution: $$2.42\cdot \sin 30°=1.33\cdot \sin r⟹\sin r=\frac{2.42\cdot 0.5}{1.33}\approx 0.909⟹r=65.4°$$

Long Answer (10 marks)

2. An object is placed at the bottom of a tank filled with water (n = 1.33) of depth 1.5 m. Calculate:
   • The apparent depth of the object.
   • The critical angle for light exiting water to air. Solution:
   • Apparent depth = $\frac{1.5}{1.33}\approx 1.13\text{ m}$.
   • Critical angle: $\sin {\theta }_c=\frac{1}{1.33}⟹{\theta }_c=48.8°$.

Conceptual (3 marks)

3. Why does a coin appear raised when viewed from the side of a bucket filled with water? Answer: Light from the coin bends away from the normal when exiting water, making the brain perceive the coin at a shallower depth.

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Key Takeaway: Refraction is all about light bending due to speed changes—master Snell’s law, apparent depth, and critical angle to ace this unit!