Refraction of light — Snell's law and total internal reflection (KCSE Physics Form 3)

TL;DR

When light passes from one transparent material to another, it bends; this bending is called refraction. Snell's Law helps us calculate exactly how much it bends, depending on the materials involved. Sometimes, light can't escape a denser material and gets trapped inside, a phenomenon known as total internal reflection.

1. The Mental Model

Imagine light as a car driving from a smooth road onto a muddy field at an angle. The wheels hitting the mud first slow down, causing the car to swerve. This swerving is like light bending when it enters a new material.

2. The Core Material

What is Refraction?

Refraction is the bending of light as it passes from one transparent medium (like air) to another (like water or glass). This happens because light changes its speed when it enters a different medium.

• When light goes from a less dense medium to a denser medium (e.g., air to water), it slows down and bends towards the normal. The normal is an imaginary line drawn perpendicular (at 90 degrees) to the surface where the light enters.
• When light goes from a denser medium to a less dense medium (e.g., water to air), it speeds up and bends away from the normal.

Refractive Index (n)

The refractive index is a measure of how much a medium can bend light. It's a ratio, so it has no units.

• Absolute refractive index (n): This is the refractive index of a medium compared to a vacuum (or air, for practical purposes).
  • n = (speed of light in vacuum) / (speed of light in medium)
  • Since the speed of light in a vacuum is the fastest, the refractive index of any other medium is always greater than 1.
• Relative refractive index: This is the refractive index of one medium compared to another.

Snell's Law

Snell's Law gives us a mathematical relationship to calculate the angles involved in refraction. It states that for a given pair of media, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant.

The formula for Snell's Law is:
n₁ sin θ₁ = n₂ sin θ₂

Where:
* n₁ is the refractive index of the first medium (where the light is coming from).
* θ₁ (theta one) is the angle of incidence. This is the angle between the incident ray and the normal.
* n₂ is the refractive index of the second medium (where the light is going into).
* θ₂ (theta two) is the angle of refraction. This is the angle between the refracted ray and the normal.

Important Note: Always measure angles from the normal, not from the surface!

Total Internal Reflection (TIR)

Total internal reflection is a special phenomenon that occurs when light tries to move from a denser medium to a less dense medium (e.g., from water to air).

Here's how it works:
1. As the angle of incidence (θ₁) in the denser medium increases, the angle of refraction (θ₂) in the less dense medium also increases.
2. At a certain angle of incidence, called the critical angle (c), the angle of refraction becomes 90 degrees. This means the refracted ray travels along the boundary between the two media.
3. If the angle of incidence is greater than the critical angle, the light ray does not refract out of the denser medium at all. Instead, it is completely reflected back into the denser medium. This is total internal reflection.

Conditions for Total Internal Reflection:
1. Light must be travelling from a denser medium to a less dense medium.
2. The angle of incidence must be greater than the critical angle.

Calculating the Critical Angle (c)

You can find the critical angle using Snell's Law. When θ₁ = c, then θ₂ = 90°.
So, n₁ sin c = n₂ sin 90°
Since sin 90° = 1, the equation becomes:
n₁ sin c = n₂
sin c = n₂ / n₁

Where:
* n₁ is the refractive index of the denser medium.
* n₂ is the refractive index of the less dense medium.

For light going from a medium (n₁) to air (n₂ ≈ 1), the formula simplifies to:
sin c = 1 / n₁

graph TD
    A[Light travels from Denser Medium to Less Dense Medium] --> B{Angle of Incidence (θ₁) ?}

    B -- θ₁ < Critical Angle (c) --> C[Refraction occurs: Light bends AWAY from Normal]
    B -- θ₁ = Critical Angle (c) --> D[Refracted ray travels ALONG the boundary (θ₂ = 90°)]
    B -- θ₁ > Critical Angle (c) --> E[Total Internal Reflection (TIR) occurs: Light reflects back into Denser Medium]

3. Worked Example

A ray of light travels from glass (refractive index n_glass = 1.5) into air (refractive index n_air = 1.0).
a) Calculate the critical angle for this glass-air interface.
b) If the angle of incidence in the glass is 40°, calculate the angle of refraction in the air.

Solution:

a) Calculating the critical angle (c):
We use the formula: sin c = n_air / n_glass
n_air = 1.0 (less dense medium)
n_glass = 1.5 (denser medium)

sin c = 1.0 / 1.5
sin c = 0.6667
c = arcsin(0.6667)
c ≈ 41.8°

b) Calculating the angle of refraction (θ₂):
We use Snell's Law: n₁ sin θ₁ = n₂ sin θ₂
Here, n₁ = n_glass = 1.5 (denser medium)
θ₁ = 40° (angle of incidence in glass)
n₂ = n_air = 1.0 (less dense medium)
θ₂ = ? (angle of refraction in air)

1.5 * sin 40° = 1.0 * sin θ₂
1.5 * 0.6428 = sin θ₂
0.9642 = sin θ₂
θ₂ = arcsin(0.9642)
θ₂ ≈ 74.6°

Since the angle of incidence (40°) is less than the critical angle (41.8°), refraction occurs, and the light ray passes into the air. If the angle of incidence had been, say, 45°, then total internal reflection would have occurred.

4. Key Takeaways

• Refraction is the bending of light as it passes from one transparent medium to another due to a change in speed.
• Light bends towards the normal when entering a denser medium and away from the normal when entering a less dense medium.
• Snell's Law (n₁ sin θ₁ = n₂ sin θ₂) quantifies the relationship between refractive indices and angles of incidence/refraction.
• Total internal reflection (TIR) occurs when light travels from a denser to a less dense medium and the angle of incidence exceeds the critical angle.
• The critical angle (c) is the angle of incidence in the denser medium where the angle of refraction is 90°.

Common Mistakes to Avoid:
* Measuring angles from the surface instead of the normal: Always use the normal as your reference line for angles.
* Confusing denser/less dense with refractive index: A higher refractive index means a denser optical medium.
* Applying TIR conditions incorrectly: Remember, TIR only happens when going from denser to less dense, and the angle of incidence must be greater than the critical angle.
* Incorrectly identifying n₁ and n₂ in Snell's Law: n₁ is always the refractive index of the medium the light is coming from, and n₂ is the medium it's going into.

5. Now Try It

A light ray passes from water (refractive index = 1.33) into diamond (refractive index = 2.42).
a) Which medium is optically denser?
b) If the angle of incidence in water is 30°, calculate the angle of refraction in diamond.
c) Would total internal reflection be possible if the light was travelling from diamond to water? Explain why or why not, and if yes, calculate the critical angle.

What success looks like:
You should be able to correctly identify the denser medium, apply Snell's Law to find the angle of refraction, and then correctly determine if TIR is possible for the reverse path, calculating the critical angle if applicable. Your answers should include the correct units (degrees for angles) and show your working clearly.