Snell's Law (AQA A Level Physics)

• Snell’s law relates the angle of incidence to the angle of refraction at a boundary between two media and 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

Critical Angle (θ_c)

• The larger the refractive index of a material, the smaller the critical angle

  • For a larger n then θ_C is smaller

• When light is shone at a boundary between a denser and a less dense material, different angles of incidence result in different angles of refraction

  • As the angle of incidence is increased, the angle of refraction also increases

  • Until the angle of incidence reaches the critical angle

• When the angle of incidence = critical angle then:

  • Angle of refraction = 90°

  • The refracted ray is refracted along the boundary between the two materials

• When the angle of incidence < critical angle then:

  • the ray is refracted and exits the material

• When the angle of incidence > critical angle then:

  • the ray undergoes total internal reflection

As the angle of incidence increases it will eventually exceed the critical angle and lead to the total internal reflection of the light

Critical Angle Equation

• The critical angle of material 1 is found using the equation:

• Where:
  • n₁ = refractive index of material 1
  • n₂ = refractive index of material 2
  • θ_c = critical angle of material 1
• The formula finds the critical angle of the denominator of the fraction
• The critical angle can also be calculated using the angles of incidence and refraction:

Total Internal Reflection

• Total internal reflection is a special case of refraction that occurs when:

  • The angle of incidence within the denser medium is greater than the critical angle (I > θ_c)

  • The incident refractive index n₁ is greater than the refractive index of the material at the boundary n₂ (n₁ > n₂)

• Total internal reflection follows the law of reflection

angle of incidence = angle of reflection

• A denser medium has a higher refractive index

  • For example, the refractive index of glass, n_g > the refractive index of air, n_a

• Light rays inside a material with a higher refractive index are more likely to be totally internally reflected

Angles of incidence, reflection and refraction to satisfy the conditions for total internal reflection

(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 
• 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