Question

Show that for a material with refractive index `µ ≥ sqrt(2)`, light incident at any angle shall be guided along a length perpendicular to the incident face.

Answer 1

Let the ray incident on face AB at angle i, after refraction, it travels along PQ and then interact with face AC which is perpendicular to the incident face.

`u = 1/(sin  i_c)`  .....(Snell's law at c)

`Sin i_c  1/mu [sin i_c = 1/mu]`

`sin(90 - r) ≥ 1/mu`  ......[∵ r + 90° + i_c = 180]

`cos r ≥ 1/mu`

`cos^2 r ≥ 1/mu^2`   ......(Squaring both sides)

`- cos^2 r ≤ 1/mu^2`

`1 - cos^2r ≤ 1 - 1/mu^2`

`sin^2r ≤ 1 - 1/mu^2`  ......(I)

`sin i/sinr = mu`  .....(By snell's law)

`sin i = mu sin r`

Or `sin^2i = mu^2 sin^2r`  ......(Squaring both sides)

`1/mu^2 sin^2i = sin^2r`  ......(II)

Put (II) in (I)

`1/mu^2 sin^2i ≤ 1 - 1/mu^2`

`sin^2i ≤ mu^2 - 1`  ......[On multiplying by μ² on both sides]

For smallest angle i.e., i = 90°

∴ `sin^2 90° ≤ mu^2 - 1`  ......`[∵ mu ≤ sqrt(2)]`

`1 + 1 ≤ mu^2`

`2 ≤ mu^2`

Taking square root

`sqrt(2) ≤ mu` Hence proved.