Question

A tank holds a layer of oil, 1.95 m thick, which floats on a layer of syrup that is 0.64 m thick. Both liquids are clear and do not intermix. A ray, which originates at the bottom of the tank on a vertical axis, crosses the oil-syrup interface at a point 0.90 m from the axis. The ray continues and arrives at the oil-air interface, 2.00 m from the axis and at the critical angle. In Figure 23.3a, the index of refraction of the oil is closest to:A) 2.00 B) 2.04 C) 2.08 D) 2.02 E) 2.06

show me drawing and work step by step

Answer 1

Option (B) 2.04, is the correct answer.

Explanation -

As mentioned in the problem, horizontal displacement of the ray = 2.0m - 0.9m = 1.1m

The Vertical displacement is the thickness of the oil, which is 1.95 m.

The angle of this beam = Arctan( 1.1m / 1.95m )
= 29.43 º

This is the critical angle for the oil/air interface. θc = 29.43º.


θc = Arsin( (index of air) / (index of oil) )

θc = Arsin( 1 / (index of oil) )

sinθc = sin[ Arsin( 1 / (index of oil) ) ]

sinθc = 1 / (index of oil)

sin(29.43º) = 1 / (index of oil)

0.4913 = 1 / (index of oil)

=> index of oil = 1 / 0.4913 = 2.035 2.04 (Answer)