A prism of angle 75^ and refractive index 3 is coated with thin...

A prism of angle

75^{\circ}

and refractive index

\sqrt{3}

is coated with thin film of refractive index 1.5 only at the back exit surface. To have total internal reflection at the back exit surface the incident angle must be

\_\_\_\_

(

\sin 15^{\circ}=0.25

and

\sin 25^{\circ}=0.43

) $A prism of angle 75^ and refractive index 3 is coated with thin film of refractive index 1.5 only at the back exit$

Options

A $15^{\circ}$
B $>25^{\circ}$
Cbetween $15^{\circ}$ and $20^{\circ}$
DBoth (1) & (3)

Correct answer

D. Both (1) & (3)

Step-by-step solution

$A prism of angle 75^ and refractive index 3 is coated with thin film of refractive index 1.5 only at the back exit$

r_1 + r_2 = 75^{\circ}

For T I R at back surface

\sqrt{3} \sin r_2 = \dfrac{3}{2} \sin 90^{\circ}

r_2 \geq 60^{\circ}

r_1 \leq 15^{\circ}

1 \cdot \sin i = \sqrt{3} \sin 15^{\circ}

\sin i = 1.73 \times 0.25

\sin i = 0.433

i = 25^{\circ} \Rightarrow i < 25^{\circ}

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