Question

A ray of light from a denser medium strikes a rarer medium at angle of incidence i. The reflected and refracted rays make an angle of 90° with each other. The angles of reflection and refraction are r & r′ respectively. The critical angle is

• A. sin^-1 (tan r)
• B. sin^-1 (cot i)
• C. tan^-1 (Sin r)
• D. tan^-1 (sin i)

Answer 1

Answer: (A)

sin^-1 (tan r)

Answer 2

Applying Snell’s Law for refraction,

…(1)

From the given condition, r + r′ = 90

⇒ Sin r′ = Cos r …(2)

Solution of (1) and (2) yields, $\frac { \sin i } { \operatorname { Cos } r } = \frac { n _ { 2 } } { n _ { 1 } }$ …(3)

Since, at the time of total internal reflection

According to the Law for refraction;

i = r …(4)

Using (3) and (4) we obtain

$\frac { \operatorname { Sin } \mathrm { i } } { \operatorname { Cos } \mathrm { i } } = \frac { \mathrm { n } _ { 2 } } { \mathrm { n } _ { 1 } }$

⇒ tan i = $\frac { \mathrm { n } _ { 2 } } { \mathrm { n } _ { 1 } }$ ….(5)

Sin θ_c = $\frac { \mathrm { n } _ { 2 } } { \mathrm { n } _ { 1 } }$, Using (5) we obtain θ_c = sin^-1 (tan r)