How to Solve: Refraction Problems

How to Solve: Refraction Problems

Introduction

"Ever wondered why a straw looks bent in water? That’s refraction—and mastering it unlocks 5-10 marks on your physics exam, from lens problems to fiber optics!

What You Need To Know First

1. Snell’s Law Basics: Light bends when it changes speed between two media.
2. Angles of Incidence/Refraction: Measured from the normal (a line perpendicular to the surface).
3. Index of Refraction (n): A number that tells you how much a medium slows down light (e.g., n for air ≈ 1, n for glass ≈ 1.5).

Key Vocabulary

Formulas To Know

1. Snell’s Law (MEMORISE THIS)
   [
   n_1 \sin \theta_1 = n_2 \sin \theta_2
   ]
2. n₁ = Index of refraction of the first medium (e.g., air).
3. θ₁ = Angle of incidence (from the normal).
4. n₂ = Index of refraction of the second medium (e.g., water).

5. θ₂ = Angle of refraction (from the normal).

6. Speed of Light in a Medium (Given on exam sheet, but know how to use it)
   [
   v = \frac{c}{n}
   ]

7. v = Speed of light in the medium (m/s).
8. c = Speed of light in a vacuum (3 × 10⁸ m/s).

9. n = Index of refraction of the medium.

10. Critical Angle Formula (MEMORISE THIS)
    [
    \sin \theta_c = \frac{n_2}{n_1} \quad \text{(where } n_1 > n_2\text{)}
    ]

11. θ_c = Critical angle (angle of incidence where refraction stops).
12. n₁ = Higher index (e.g., glass).
13. n₂ = Lower index (e.g., air).

Step-by-Step Method

Follow these steps for every refraction problem:

1. Draw a Diagram
2. Sketch the boundary between the two media (e.g., air → water).
3. Draw the normal (a dashed line perpendicular to the surface).

4. Label the incoming ray, angle of incidence (θ₁), and refracted ray.

5. Identify Given Values

6. Write down n₁, n₂, θ₁, or θ₂ (whichever are provided).

7. If n isn’t given, check your formula sheet (e.g., n for water = 1.33).

8. Choose the Right Formula

9. Use Snell’s Law if you need to find an angle or n.
10. Use v = c/n if you need speed.

11. Use critical angle formula if the question mentions total internal reflection.

12. Plug in the Numbers

13. Substitute known values into the formula.

14. If solving for an angle, use sin⁻¹ (inverse sine) on your calculator.

15. Check Units & Reasonableness

16. Angles should be between 0° and 90°.
17. If n₁ > n₂, light bends away from the normal (e.g., glass → air).

18. If n₁ < n₂, light bends toward the normal (e.g., air → water).

19. Write the Final Answer with Units

20. Angles: degrees (°).
21. Speed: m/s.
22. Index of refraction: no units.

Worked Example Using the Steps

Problem: A light ray travels from air (n = 1.00) into glass (n = 1.52) at an angle of incidence of 30°. What is the angle of refraction?

Solution: 1. Draw a Diagram
- Air (top) → Glass (bottom).
- Normal line drawn at 90° to the surface.
- Incoming ray at 30° to the normal.

1. Identify Given Values
2. n₁ = 1.00 (air)
3. n₂ = 1.52 (glass)
4. θ₁ = 30°

5. θ₂ = ? (what we’re solving for)

6. Choose the Right Formula

7. Use Snell’s Law: n₁ sin θ₁ = n₂ sin θ₂

8. Plug in the Numbers
   [
   1.00 \times \sin 30° = 1.52 \times \sin \theta_2
   ]
   [
   0.5 = 1.52 \times \sin \theta_2
   ]
   [
   \sin \theta_2 = \frac{0.5}{1.52} = 0.3289
   ]
   [
   \theta_2 = \sin^{-1}(0.3289) = 19.2°
   ]

9. Check Reasonableness

10. Since n₂ > n₁, light bends toward the normal → angle should be smaller than 30° (19.2° makes sense).

11. Final Answer
    [
    \theta_2 = 19.2°
    ]

Worked Examples

Example 1 – Basic (Straightforward Snell’s Law)

Problem: Light passes from water (n = 1.33) into air (n = 1.00) at an angle of 25°. What is the angle of refraction?

Solution: 1. Diagram: Water → Air, normal line, 25° incidence. 2. Given: n₁ = 1.33, n₂ = 1.00, θ₁ = 25°. 3. Formula: Snell’s Law. 4. Calculation:
[
1.33 \sin 25° = 1.00 \sin \theta_2
]
[
1.33 \times 0.4226 = \sin \theta_2
]
[
\sin \theta_2 = 0.5621
]
[
\theta_2 = \sin^{-1}(0.5621) = 34.2°
] 5. Check: n₂ < n₁ → light bends away from normal → angle increases (34.2° > 25° ✔). 6. Answer: 34.2°

What we did and why: We used Snell’s Law to find the refracted angle when light moves from a slower medium (water) to a faster one (air). The angle increased because light speeds up.

Example 2 – Medium (Finding n)

Problem: A light ray enters an unknown liquid at 40° and refracts at 28°. If the first medium is air (n = 1.00), what is the index of refraction of the liquid?

Solution: 1. Diagram: Air → Liquid, normal line, 40° incidence, 28° refraction. 2. Given: n₁ = 1.00, θ₁ = 40°, θ₂ = 28°, n₂ = ? 3. Formula: Snell’s Law. 4. Calculation:
[
1.00 \sin 40° = n_2 \sin 28°
]
[
0.6428 = n_2 \times 0.4695
]
[
n_2 = \frac{0.6428}{0.4695} = 1.37
] 5. Check: n₂ > n₁ → light bends toward normal → angle decreases (28° < 40° ✔). 6. Answer: 1.37

What we did and why: We rearranged Snell’s Law to solve for n₂ (the unknown liquid’s index). The angle decreased, confirming the liquid slows light more than air.

Example 3 – Exam Style (Critical Angle)

Problem: A fiber optic cable has a core with n = 1.48 and a cladding with n = 1.46. What is the critical angle for light traveling from the core to the cladding?

Solution: 1. Diagram: Core → Cladding, normal line, critical angle (θ_c). 2. Given: n₁ = 1.48 (core), n₂ = 1.46 (cladding). 3. Formula: Critical angle formula. 4. Calculation:
[
\sin \theta_c = \frac{n_2}{n_1} = \frac{1.46}{1.48} = 0.9865
]
[
\theta_c = \sin^{-1}(0.9865) = 80.6°
] 5. Check: n₁ > n₂ → total internal reflection possible → critical angle exists (80.6° is reasonable). 6. Answer: 80.6°

What we did and why: We used the critical angle formula because the question asked for the angle where light stops refracting and reflects instead. This is key for fiber optics!

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in for your exam. Refraction is just light bending when it changes speed. Remember Snell’s Law: n₁ sin θ₁ = n₂ sin θ₂—that’s your golden ticket. Draw the normal, label your angles, and plug in the numbers. If n increases, light bends toward the normal; if n decreases, it bends away. For critical angles, use sin θ_c = n₂/n₁ when light goes from slow to fast. Watch out for examiners hiding the normal or swapping n₁ and n₂. Double-check your calculator is in degrees, not radians! You’ve got this—go ace that question!

Teacher’s Notes for Recording:

• Pacing: Spend 30 sec on the hook, 1 min on prerequisites/vocab, 2 min on formulas, 3 min on step-by-step + example, 2 min on worked examples, 1 min on mistakes/traps, and 1 min on recap.
• Visuals: Show diagrams for every example. Highlight the normal line in red.
• Engagement: Ask students to pause and predict the answer before revealing it (e.g., "Will the angle increase or decrease when light goes from water to air?").
• Calculator Demo: Show how to use sin⁻¹ on screen.