Total internal reflection — critical angle derivation

What we are explaining

Derivation from first principles

WHAT we start with — Snell's law. This is the only physics input we need: $n_1 \sin\theta_1 = n_2 \sin\theta_2$

WHY this law? Light slows down in a denser medium ($v = c/n$). At a boundary the wavefronts must stay continuous, so the ray must bend. Snell's law is exactly the bookkeeping of that continuity (it follows from Fermat's least-time principle / phase matching).

HOW we find the critical angle — step by step.

Step 1. Send light from medium 1 (denser, $n_1$) into medium 2 (rarer, $n_2$), so $n_1 > n_2$. Why this step? TIR is impossible going rarer→denser, so we must set up dense→rare.

Step 2. Increase $\theta_1$. Since $n_1 > n_2$, Snell gives $\sin\theta_2 = \dfrac{n_1}{n_2}\sin\theta_1 > \sin\theta_1$, so $\theta_2 > \theta_1$: the ray bends away from the normal. Why this step? It shows $\theta_2$ races ahead of $\theta_1$ and will hit $90°$ first.

Step 3. Define the critical angle as the $\theta_1$ for which the refracted ray just grazes the surface: $\theta_2 = 90°$. Why this step? $90°$ is the last possible refraction angle. Beyond it, no real refracted ray exists.

Step 4. Substitute $\theta_1 = \theta_c$ and $\theta_2 = 90°$ into Snell: $n_1 \sin\theta_c = n_2 \sin 90° = n_2 \cdot 1$

Step 5. Solve:

Step 6 (the punchline). For $\theta_1 > \theta_c$, Snell would demand $\sin\theta_2 = \frac{n_1}{n_2}\sin\theta_1 > 1$ — impossible. So no refracted ray exists, and 100% of the energy reflects: total internal reflection.

Worked examples

Common mistakes

Flashcards

Connections

• Snell's Law — the parent law TIR is derived from.
• Refraction of Light — TIR is the limiting case of refraction.
• Refractive Index — controls the size of $\theta_c$.
• Optical Fibres — TIR keeps light trapped along the fibre core.
• Mirage and Atmospheric Refraction — gradual TIR from warm-air layers.
• Brewster's Angle — contrast: polarization angle, not refraction-ceasing angle.
• Prisms and Total Internal Reflection — periscopes/binoculars use $45°>\theta_c$ in glass.

Concept Map

Hinglish (regional understanding)