2B.5 – Critical Angle | Physics with Mr Shone

Consider a light ray travelling from an optically denser medium (of refractive index n) to vacuum (or air) (optically less dense).

θ₁ is smaller than θ₂. If the angle of incidence θ₁ of the incident ray is gradually increased, the angle of refraction θ₂would increase and reach 90° At this point, let θ₁ = θ_C.

We call angle c the critical angle.

Definition: Critical Angle
Critical angle c is defined as the angle of incidence in an optically denser medium for which the angle of refraction in the less dense medium is 90º.

It is related to refractive index, n, by the relationship:

n = 1 / sin c

Note: this relationship only applies when light is travelling from a medium into vacuum or air

Therefore, the critical angle is formally defined as the minimum angle of incidence in the medium with higher refractive index (n) at which a light ray is totally reflected.
The phenomenon where light is completely reflected is known as total internal reflection (TIR).

Example 5
Consider a light ray travelling from an optically denser medium (refractive index n) to air at the critical angle c. Apply Snell’s law to show that n = 1 / sin c Applying Snell’s law, n₁sin θ₁ = n₂sin θ₂ n₁ = n θ₁ = c n₂ = 1.0 θ₂ = 90^o Hence, n × sin c = 1.0 × sin 90° = 1 n = 1 / sin c

Example 5

Consider a light ray travelling from an optically denser medium (refractive index n) to air at the critical angle c.

Apply Snell’s law to show that n = 1 / sin c

Applying Snell’s law,

n₁sin θ₁ = n₂sin θ₂

n₁ = n

θ₁ = c

n₂ = 1.0

θ₂ = 90^o

Hence,

n × sin c = 1.0 × sin 90° = 1

n = 1 / sin c

Example 6
The refractive indices of glass and air are 1.5 and 1.0 respectively. A ray of light travels from glass into air. (a) State the angle of refraction in air when the angle of incidence in glass is equal to the critical angle. 90º (b) Calculate c, the critical angle of the glass-air boundary. Using n = 1/sin c sin c =1/n = 1/1.5 c = 41.8º

Example 6

The refractive indices of glass and air are 1.5 and 1.0 respectively. A ray of light travels from glass into air.

(a) State the angle of refraction in air when the angle of incidence in glass is equal to the critical angle.

90º

(b) Calculate c, the critical angle of the glass-air boundary.

Using n = 1/sin c

sin c =1/n = 1/1.5

c = 41.8º

Enrichment
However, it has been observed that at critical angle, the refracted ray is either faint or cannot be seen. This is due to two factors: Light is partially reflected during refraction Intensity of the refracted ray changes based on the angle of incidence Generally, when the angle of incidence increases, the intensity of the refracted ray will decrease and the intensity of the reflected ray will increase. As the angle of incidenceapproaches the critical angle, the intensity of the refracted ray gradually decrease to zero. This implies that the incident ray is now entirely reflected.

Enrichment

However, it has been observed that at critical angle, the refracted ray is either faint or cannot be seen.

This is due to two factors:

Light is partially reflected during refraction
Intensity of the refracted ray changes based on the angle of incidence

Generally, when the angle of incidence increases, the intensity of the refracted ray will decrease and the intensity of the reflected ray will increase.
As the angle of incidenceapproaches the critical angle, the intensity of the refracted ray gradually decrease to zero. This implies that the incident ray is now entirely reflected.

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