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

$n_{1} sin(\theta_{p}) = n_{2}cos(\theta_{p})$
$\frac{sin(\theta_{p}}{cos(\theta_{p})} = \frac{n_{2}}{n_{1}}$
$tan(\theta_{p}) = \frac{n_{2}}{n_{1}}$      (2)

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

$tan(\theta_{p}) = \frac{n_{1} sin(\theta_{c})}{n_{1}}$
$tan(\theta_{p}) = sin(\theta_{c})$

Substituting the value of the critical angle, we have

$tan(\theta_{p}) = sin(42)$
$\theta_{p} = tan^{-1}(sin42) = 33.79$

Answer: 33.79°