Critical angle

When light passes from one medium (material) to another it changes speed. This is because the speed of a wave is determined by the medium through which it is passing.

When light speeds up as it passes from one material to another, the angle of refraction is bigger than the angle of incidence.

For example, this happens when light passes from water to air or from glass to water.

Angle of incidence and angle of refraction through water and air

The diagram above shows light incident on a water-air interface.

  • angle of incidence is the angle between an incident ray and the normal
  • angle of refraction is the angle between a refracted ray and the normal

When the angle of refraction is equal to 90^\circ, the angle of incidence is called the critical angle, \theta_{c}.

The angle of refraction cannot be greater than 90^\circ. Look at the two images below to see what happens as the angle of incidence increases.

Angle of refraction less than 90 degrees

Angle of incidence less than critical angle

As the angle of incidence increases, the angle of refraction gets closer to ninety degrees.

At any angle of incidence greater than the critical angle, the light cannot pass through the surface - it is all reflected.

This is called total internal reflection.

  • Total because all of the energy is reflected
  • Internal because the energy stays inside the material
  • Reflection because the light is reflected

The relationship between critical angle, \theta_{c}, and refractive index, n is sin\theta _{c}=\frac{1}{n}.

Question

Calculate the critical angle for red light incident on a water-air interface.

The refractive index of water is 1.33 for this colour of light.

n= 1.33

sin\theta _{c}=\frac{1}{n}

sin\theta _{c}= \frac{1}{1.33}

\theta _{c}=48.7535^\circ

Critical angle of water for this light =48.8^\circ

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