# A glass prism of refractive index 1.5 is immersed in water of refractive index 1.33. A ray of light is incident normally on face AB is totally reflected. find the limiting angle for the phenomenon?

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Look at the diagram enclosed. ABC is a glass prism with some angles A, B, and C.

__Critical angle for total internal reflection at Glass - water interface = Ф. __

Sin Ф = 1.33 / 1.50 => Ф = 62.5°

The question does not specifically say whether total reflection takes place only at face AC, or at AC and BC. Further it does not say the angle at which the emergent ray makes. It does not say from which face the ray emerges.

I have considered various possibilities. R2, R3, R4, and R5 are reflected rays.

__Case 1)__

for the light ray R1 to be totally internally reflected at face AC, angle of incidence = A, must be more than Ф = 62.5°.

__case 2)__

For the reflected ray DE , at face CB, angle of incidence = C - A.

C - A > Ф for total internal reflection at CB.

Hence , C > A + Ф ie., C > 2 Ф

Thus we get* ***3 Ф < A+ C < 180° ** ie., ** Ф < 60°**

This is not true in our case, as Ф = 62.5°. So The Ray DE will refract out of prism through face CB.

The condition for the ray to go out from face CB:

0 < C - A < Ф , ie., Ф < A < C < A + Ф

62.5° < A < C < A + 62.5°

==========================================

The following cases are valid in case critical angle Ф is less than 60°.

__case 3)__

consider the quadrilateral FDEG : and ray R2.

angles 90° + 2 A + 2 (C - A) + (90° + i) = 360°

i = 180 - 2 C

For total internal reflection, and R2 to pass thru face AB, i < Ф

**angle C > 90 - Ф/2**

case 4)

consider the ray R3 , which is parallel to R1 incident ray.

angles in quadrilateral FDEH = 90 + 2 A + 2 (C - A) + 90 = 360°

=>**C = 90 °**

**Case 5) **

*Consider the ray R4 *, which is refracted on the surface AB.

angles in quadrilateral FDEJ = 90 + 2 A + 2 (C-A) + (90 - i) = 360°

=> i > 0 and i = 2 C - 180 < Ф for refraction only.

** 90° < C < 90 + Ф/2**

The question does not specifically say whether total reflection takes place only at face AC, or at AC and BC. Further it does not say the angle at which the emergent ray makes. It does not say from which face the ray emerges.

I have considered various possibilities. R2, R3, R4, and R5 are reflected rays.

for the light ray R1 to be totally internally reflected at face AC, angle of incidence = A, must be more than Ф = 62.5°.

For the reflected ray DE , at face CB, angle of incidence = C - A.

C - A > Ф for total internal reflection at CB.

Hence , C > A + Ф ie., C > 2 Ф

Thus we get

This is not true in our case, as Ф = 62.5°. So The Ray DE will refract out of prism through face CB.

The condition for the ray to go out from face CB:

0 < C - A < Ф , ie., Ф < A < C < A + Ф

62.5° < A < C < A + 62.5°

==========================================

The following cases are valid in case critical angle Ф is less than 60°.

consider the quadrilateral FDEG : and ray R2.

angles 90° + 2 A + 2 (C - A) + (90° + i) = 360°

i = 180 - 2 C

For total internal reflection, and R2 to pass thru face AB, i < Ф

case 4)

consider the ray R3 , which is parallel to R1 incident ray.

angles in quadrilateral FDEH = 90 + 2 A + 2 (C - A) + 90 = 360°

=>

angles in quadrilateral FDEJ = 90 + 2 A + 2 (C-A) + (90 - i) = 360°

=> i > 0 and i = 2 C - 180 < Ф for refraction only.