Library / Foundations: Physics / FPH-34

Foundations: Physics

Critical angle

Angle beyond which light is totally internally reflected.

Ref: FPH-34
Rev: v1
Field: Foundations: Physics
Result: θ_c [°]

\sin θ_{c} = \frac{n_{2}}{n_{1}}

At the critical angle, sin θc = n₂/n₁ for light passing from higher to lower refractive index (n₁ > n₂).

If light inside a dense medium hits the boundary too shallowly, it cannot escape and reflects completely.

Calculator

Solve for

θ_cCritical angleComputed

n₁Denser index

1.22.4

n₂Less dense index

11.1

sin θc0.6667
Speed of light in the denser medium1.999 10⁸ m/s

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Critical angle (°) current value 41.81 °

n₁Denser indexKey driver

24.656.4

n₂Less dense index

47.2

Glass–air has a critical angle of about 42°, which is why prisms can act as mirrors.
Optical fibres keep light inside by total internal reflection.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (°)	Local step	Change in result
n₁ Denser index	1.2 to 2.4	24.62	56.44	31.82	+undefined%	—
n₂ Less dense index	1 to 1.1	41.81	47.17	5.356	+undefined%	—

Sweep of Denser index

Denser index	Critical angle (°)
1.2	56.44
1.35	47.79
1.5	41.81
1.65	37.31
1.8	33.75
1.95	30.85
2.1	28.44
2.25	26.39
2.4	24.62

Sweep one variable

Vary

Critical angle (°)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

θ_c

Critical angle

Result · °

What the equation gives you. Everything else feeds it.

n₁

Denser index

· typical 1.2 to 2.4

Refractive index of the denser medium. A higher n₁ lowers the critical angle.

n₂

Less dense index

· typical 1 to 1.1

Refractive index of the other medium (1 for air). A higher n₂ raises the critical angle.

Worked example with the values above

sin θc = n₂/n₁ = 1 / 1.5 = 0.6667.
Inverse sine.
θ_c = 41.81 °.

Assumptions and limits

Light travels from the denser to the less dense medium.
Smooth boundary.

Common mistakes

Putting the indices the wrong way round.
Measuring the angle from the surface.

Statistics and uncertainty

A 1% error in either refractive index changes the critical angle by under 1°.

Where it is used

Optical fibre and light-guide design.
Prism reflectors.

Maths you need

Snell’s law: refraction
Arcsine: inverse sine

Related formulas

Practise this formula