A rectangular glass slab ABCD of refractive index $n_{1}$ is immersed in water of refractive index $n_{2}$ $(n_{1} > n_{2})$ . A ray of light is incident at the surface AB of the slab as shown. The maximum value of the angle of incidence $α_{\max}$ such that the ray comes out only from the other surface CD is given by [2000]

Question

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(1)

From the diagram, the ray will come out from CD if it suffers total internal reflection at surface AD, i.e., it strikes the surface AD at critical angle $θ_{c}$ (the limiting case).

Applying Snell's law at P face AD of slab,

$n_{1} \sin θ_{c} = n_{2}$ or $\sin θ_{c} = \frac{n_{2}}{n_{1}}$

Applying Snell's law at Q,

$n_{2} \sin α = n_{1} \cos θ_{c}$

$\Rightarrow \sin α = \frac{n_{1}}{n_{2}} \cos {\sin^{- 1} (\frac{n_{2}}{n_{1}})}$

or $α = \sin^{- 1} [\frac{n_{1}}{n_{2}} \cos {\sin^{- 1} (\frac{n_{2}}{n_{1}})}]$

Solution

1 $\sin^{- 1} [\frac{n_{1}}{n_{2}} \cos (\sin^{- 1} (\frac{n_{2}}{n_{1}}))]$

2 $\sin^{- 1} [n_{1} \cos (\sin^{- 1} (\frac{1}{n_{2}}))]$

3 $\sin^{- 1} (\frac{n_{1}}{n_{2}})$

4 $\sin^{- 1} (\frac{n_{2}}{n_{1}})$

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Solution

1 sin-1[n1n2cos(sin-1(n2n1))]

2 sin-1[n1cos(sin-1(1n2))]

3 sin-1(n1n2)

4 sin-1(n2n1)

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1 $\sin^{- 1} [\frac{n_{1}}{n_{2}} \cos (\sin^{- 1} (\frac{n_{2}}{n_{1}}))]$

2 $\sin^{- 1} [n_{1} \cos (\sin^{- 1} (\frac{1}{n_{2}}))]$

3 $\sin^{- 1} (\frac{n_{1}}{n_{2}})$

4 $\sin^{- 1} (\frac{n_{2}}{n_{1}})$