Question

Find the value of θ in the given figure.

Options

• ${\displaystyle {\sin }^{-1}\left(\frac{2}{\sqrt{3}}\right)}$

• ${\displaystyle {\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)}$

• ${\displaystyle {\sin }^{-1}\left(\frac{1}{2}\right)}$

• ${\displaystyle {\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)}$

Answer 1

${\displaystyle {\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)}$

Explanation:

For total internal reflection, μ = ${\displaystyle \frac{1}{\sin \text{c}}}$​

where, C = critical angle

Condition for TIR, the angle of incidence must be greater than critical angle C,

(90 - θ₁) > C

sin (90 - θ₁) > sinC

cosθ₁ = ${\displaystyle \frac{1}{\mu }}$

cosθ₁ > ${\displaystyle \frac{1}{2/\sqrt{3}}}$

cosθ₁ > ${\displaystyle \frac{\sqrt{3}}{2}}$

θ₁ > 30°

According to Snell’s law,

μ₁sinθ = μ₂sinθ₁

1sinθ = ${\displaystyle \frac{2}{\sqrt{3}}}$sinθ₁

sinθ = ${\displaystyle \frac{2}{\sqrt{3}}}$sin30°

sinθ = ${\displaystyle \frac{1}{\sqrt{3}}}$

So, θ = ${\displaystyle {\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)}$