Total internal reflection — critical angle derivation: Foundations

Before you can derive the critical angle, you must own every letter and picture the parent note throws at you. This page builds each one from nothing, in the order they depend on each other.

1. The boundary and the two media

The whole story happens at a flat surface where two see-through materials meet — say glass sitting under air, or water under air.

Figure — Total internal reflection — critical angle derivation

Look at the figure: the horizontal chalk line is the boundary (also called the interface). Medium 1 fills the bottom, Medium 2 the top. A single ray of light climbs up and hits the boundary at one point.

2. The normal — the reference line for all angles

You cannot say "the angle of the ray" without saying angle from what. In optics we never measure angles from the surface itself. We measure from a special line called the normal.

In the figure the normal is the vertical dashed chalk line. Notice the little square corner symbol — that marks the $90°$ between the normal and the surface.

3. Incident ray, refracted ray, and their angles $θ_{1}, θ_{2}$

Now that we have a reference line, we can name the angles.

Study this figure carefully — it is the whole geometry of refraction on one board:

The incident ray (chalk blue) comes up from the bottom, hitting the surface.
$θ_{1}$ is the pale-yellow angle between it and the normal.
The refracted ray (chalk pink) continues into Medium 2, bent.
$θ_{2}$ is the angle between that and the normal.

Recall Quick self-check on the picture

If a ray goes straight up along the normal, what are $θ_{1}$ and $θ_{2}$ ? ::: Both $0°$ — the ray is on the normal, so there is no gap between ray and normal, and it passes straight through with no bending.

4. Refractive index $n$ — the "slowness" number

Why does light bend at all? Because it changes speed. Each medium gets a number that captures how much it slows light down.

Two labels we'll reuse constantly:

See Refractive Index for the full story of where $n$ comes from; here we only need "bigger $n$ = slower light".

5. What " $sin$ " really means (built from a triangle)

Before we can write Snell's Law we must first meet the one piece of mathematics it leans on: the sine of an angle. You should never use a symbol you cannot picture, so we build $sin θ$ from zero before it appears in any law.

The figure shows a right triangle with the angle $θ$ marked. The pink side is opposite $θ$ ; the long slanted side is the hypotenuse. As $θ$ grows, the opposite side grows toward the hypotenuse's full length, so the ratio climbs toward $1$ .

Intuition Why sine is capped at 1 — and why that cap IS the whole topic

The opposite side can never be longer than the hypotenuse (the hypotenuse is always the longest side of a right triangle). So $sin θ$ can never exceed 1. Its biggest possible value is exactly $1$ , reached when $θ = 90°$ .

This single fact is the engine of total internal reflection (the phenomenon where light bounces entirely back into the denser medium — spelled out fully in Section 8 and named there). Snell's law, which we meet next, will say $sin θ_{2} = \frac{n _{1}}{n _{2}} sin θ_{1}$ . Going denser→rarer, $\frac{n _{1}}{n _{2}} > 1$ , so this can demand a value bigger than 1 — which is geometrically impossible. When the maths asks for the impossible, refraction quits and the light reflects. Everything on the parent page is spelling out this sentence.

Recall Sine sanity values

$sin 0° = ?$ ::: $0$ (the opposite side has zero length — ray lies on the normal). $sin 90° = ?$ ::: $1$ (opposite side equals the hypotenuse — ray lies flat along the surface). $sin 30° = ?$ ::: $0.5$ (a handy value used in Example 3 of the parent).

6. Snell's Law — the one physics rule that ties it together

Now that we can picture $sin θ$ , we can connect the two angles ( $θ_{1}, θ_{2}$ ) using the two indices ( $n_{1}, n_{2}$ ).

Read the Snell's Law note for the derivation; on this page we treat it as our single given input, exactly as the parent does.

7. Inverse sine $sin^{- 1}$ — undoing the sine

Snell's law hands us a number for a sine; to report an actual angle we must run sine backwards.

8. The critical angle $θ_{c}$ — the symbol the whole page is chasing

Everything above now lets us state the payoff symbol precisely — including the exact conditions under which it even exists.

This is why the domain condition matters: the formula $sin θ_{c} = n_{2} / n_{1}$ only gives a value $\leq 1$ (a valid sine) when $n_{2} < n_{1}$ . If you ever plug in $n_{1} < n_{2}$ and get " $sin θ_{c} > 1$ ", that is the maths telling you no critical angle exists.

With this in hand, the parent's derivation reads cleanly: set $θ_{2} = 90°$ , use $sin 90° = 1$ , and Snell collapses to $sin θ_{c} = n_{2} / n_{1}$ ; then apply $sin^{- 1}$ (principal branch) to read off the angle.

Prerequisite map

Equipment checklist

Cover the right side; can you answer each before moving to the derivation page?

What is the "normal" and why do we measure angles from it, not the surface?

A line perpendicular to the boundary at the hit-point; measuring from it makes a straight-through ray read as

0°

, keeping the easy case easy.

What do

θ_{1}

and

θ_{2}

stand for?

The angle of incidence and the angle of refraction, each measured between its ray and the normal.

Define the refractive index

n

in one line.

n = c / v

— how many times slower light moves in the medium than in vacuum; always

\geq 1

Which has the higher

n

, a denser or a rarer medium?

The denser medium (light is slower there).

What is

sin θ

in terms of a triangle, and what is its maximum value?

Opposite side over hypotenuse; maximum is

1

(at

θ = 90°

) because the opposite side can't beat the hypotenuse.

Why is that maximum-of-1 the key to total internal reflection?

Going denser→rarer, Snell can demand

sin θ_{2} > 1

, which is impossible, so refraction stops and light reflects fully.

State Snell's law and what each symbol is.

n_{1} sin θ_{1} = n_{2} sin θ_{2}

; indices

n_{1}, n_{2}

and angles

θ_{1}, θ_{2}

on the two sides of the boundary.

What does

sin^{- 1} (x)

do, and which angle does it return?

Returns the angle whose sine is

x

, chosen from the principal branch

[- 90°, 90°]

— converting a ratio back into degrees.

Define the critical angle

θ_{c}

, including when it exists.

The incidence angle at which the refracted ray grazes the surface (

θ_{2} = 90°

); it exists only when

n_{1} > n_{2}

(denser→rarer).

Connections

Snell's Law — the single physics law these foundations feed into.
Refractive Index — full story of the number $n$ used above.
Refraction of Light — the bending these angles describe.
Total internal reflection — critical angle derivation — the parent topic this page prepares you for.