# 4.8.2 Total Internal Reflection

### Total Internal Reflection

• 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 C
• This angle can be found using the formula: • This can easily be derived from Snell’s law where:
• θ1 = C
• θ2 = 90°
• nn
• n2 = 1 (air)
• 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, θ1 > the critical angle, C
• Refractive index n1 > refractive index n2 (air) #### Worked Example

A glass cube is held in contact with a liquid and a light ray is directed at a vertical face of the cube.

The angle of incidence at the vertical face is 39° and the angle of refraction is 25° as shown in the diagram.

The light ray is totally internally reflected at X. Complete the diagram to show the path of the ray beyond X to the air and calculate the critical angle for the glass-liquid boundary. 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 (and 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
• Due to the reflection, the light rays are symmetrical to the other side

Step 3: Calculate the critical angle

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

#### Exam Tip

Always draw ray diagrams with a ruler, and make sure you’re comfortable calculating unknown angles. The main rules to remember are:

• Angles in a right angle add up to 90°
• Angles on a straight line add up to 180°
• Angles in any triangle add up to 180°

For angles in parallel lines, such as alternate and opposite angles, take a look at the OCR GCSE maths revision notes ‘7.1.1 Angles in Parallel Lines’

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