Question

# A slab of refractive index $$\mu$$ is placed in air and light is incident at maximum angle $$\theta _{0}$$ from vertical. Find minimum value of $$\mu$$ for which total internal reflection takes place at the vertical surface.

Solution

## Using Snell’s law at the surface S1: $$\mu \sin r=\sin {{\theta }_{0}}............(1)$$ For TIR at vertical surface:   $$\mu \sin {{\theta }_{c}}=\sin 90$$  $$\mu \sin {{\theta }_{c}}=1$$  $$\because {{\theta }_{c}}=90-r$$  $$\mu \sin (90-r)=1$$  $$\mu \cos r=1......(2)$$ Solving equation (1) and (2)   $${{\mu }^{2}}({{\sin }^{2}}r+{{\cos }^{2}}r)={{\sin }^{2}}{{\theta }_{0}}+1$$  $${{\mu }^{2}}={{\sin }^{2}}{{\theta }_{0}}+1$$  $$\mu =\sqrt{{{\sin }^{2}}{{\theta }_{0}}+1}$$  Physics

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