Question

The critical angle for total internal reflection for a substance is 42.0°. what is the polarizing angle for this substance?

Answer 1

For the total internal reflection of a substance, we can express the relationship between the critical angle,
`\theta_(c)`, and the indices of refraction as


`n_(1) sin(\theta_(c)) = n_(2)` (1)

where
`n_(1)` and
`n_(2)` are the indices of refraction. And under polarization, we can express Snell's Law as


`n_(1) sin(\theta_(p)) = n_(2)sin(90-\theta_(p))`

Rearranging this we have


`n_(1) sin(\theta_(p)) = n_(2)cos(\theta_(p))`

`(sin(\theta_(p))/(cos(\theta_(p))) = (n_(2))/(n_(1))`

`tan(\theta_(p)) = (n_(2))/(n_(1))` (2)

Substituting equations (1) into (2), we have


`tan(\theta_(p)) = (n_(1) sin(\theta_(c)))/(n_(1))`

`tan(\theta_(p)) = sin(\theta_(c))`

Substituting the value of the critical angle, we have


`tan(\theta_(p)) = sin(42)`

`\theta_(p) = tan^(-1)(sin42) = 33.79`

Answer: 33.79°