3.5.2 Snell's Law

• Snell’s law relates the angle of incidence to the angle of refraction, it is given by:

n₁ sin θ₁ = n₂ sin θ₂

• Where:
  • n₁ = the refractive index of material 1
  • n₂ = the refractive index of material 2
  • θ₁ = the angle of incidence of the ray in material 1
  • θ₂ = the angle of refraction of the ray in material 2

• θ₁ and θ₂ are always taken from the normal
• Material 1 is always the material in which the ray goes through first
• Material 2 is always the material in which the ray goes through second

• As the angle of incidence is increased, the angle of refraction also increases until it gets closer to 90°
• When the angle of refraction is exactly 90° the light is refracted along the boundary
  • At this point, the angle of incidence is known as the critical angle θ_c

• This angle can be found using the formula:

• This can easily be derived from Snell’s law where:
  • θ₁ = θ_c
  • θ₂ = 90°
  • n₁ > n₂

• Total internal reflection (TIR) occurs when:

The angle of incidence is greater than the critical angle and the incident refractive index n₁ is greater than the refractive index of the material at the boundary n₂

• Therefore, the two conditions for total internal reflection are:
  • The angle of incidence > the critical angle
  • The refractive index n₁ is greater than the refractive index n₂

(i)Step 1: Draw the reflected angle at the glass-liquid boundary

• When a light ray is reflected, the angle of incidence = angle of reflection
• Therefore, the angle of incidence (or reflection) is 90° – 25° = 65°

Step 2: Draw the refracted angle at the glass-air boundary

• At the glass-air boundary, the light ray refracts away from the normal when the ray leaves to air (as air is less dense)
• Due to the reflection, the light rays are symmetrical on either side of the glass (39^o)

Step 3: Calculate the critical angle for the glass-liquid boundary

• The question states the ray is “totally internally reflected for the first time” meaning that this is the smallest angle at which TIR occurs 
• Therefore, 65° is the critical angle

(ii)Step 1: Write down the known quantities

• Refractive index of glass, n₁ = 1.489 (from previous worked example)
• Refractive index of liquid = n₂
• Angle of incidence / critical angle, θ₁ = θ_c = 65°
• Angle of refraction, θ₂ = 90°

Step 2: Write out Snell’s Law or the equation for critical angle

Step 3: Calculate the refractive index of the liquid