Light enters at an angle of incidence in a transparent rod of refractive index n. For what value of the refractive index of the material of the rod the light once entered into it will not leave it through its lateral face what so ever be the value of angle of incidence?

Let a ray of light enter at A and the refracted beam is AB. This is incident at angle θ . For no refraction at the lateral face, θ>C  or,  sinθ>sinC But θ+r=900⇒θ=900−r∴sin900−r>sinC Or   cosr>sinC..................................iFrom Snell’s law,  n=sinisinr⇒sinr=sinin ∴cosr=1−sin2r=1−sin2in2∴ equation (i) we get  1−sin2in2>sinC⇒1−sin2in2>sin2CAlso,  sinC=1n∴  1−sin2in2>1n2 or 1>sin2in2+1n2Or 1n2sin2i+1<1 or  n2>sin2i+1Maximum value of  sini=1 Hence   n2 > 2 ⇒n > 2