Question

A point source of light is placed at a distance h below the surface of a large and deep lake. Show that the fraction f of the light energy that escapes directly from the water surface is independent of n and is given by 

(a) f = 1/2 [1- √1- (1/n)²] 

(b) f = 1/2 [1+ √1- (1/n)²] 

(c) f = 1/2 [1- √1+ (1/n)²] 

(d) f = 1/2 [1+ √1+ (1/n)²]  

Answer 1

Correct Option :- (a) f = 1 2 [1- √1- (1/n)2] 

Explanation :-

As due to total internal reflection, light will be reflected back into water if i > θ_c , so only the position of incident light will escape which pass through the core of angle θ =2θ_c .  

So the fraction of light Escaping 

f = area of cap /area of sphere 

f =2πRy/4πR² 

f = 2πy/4πR = y/2R 

f = [(1/2) (R-h)/R] 

f = (1/2)[1-(h/R)] 

f = (1/2) [1-(h/R)] 

f =(1/2) [1- cos θ_c] 

f = (1/2)[1- √1- sin2θc] 

Sin θ_c = 1/n f = (1/2) [1- √1-(1/n)²] 

f = ½[1-√1- (1/n)²]