Question
Question: The critical angle for glass is \(41^{\circ}48\prime\) and that for water is \(48^{\circ}36\prime\)....
The critical angle for glass is 41∘48′ and that for water is 48∘36′. Calculate the critical angle for glass-water interface.
A.62∘28′
B.34∘42′
C.52∘42′
D.44∘42′
Solution
Critical angle and the refractive index of the material are related bysinC=μ1=cv. Since the critical angle of two different systems is given, we can find the refractive index of each system and then the refractive index of the combined materials. This refractive index will give the required critical angle.
Formula used:
sinC=μ1=cv
Complete step-by-step answer:
We know that the angle of incidence is called the critical angle Cif the angle of the refracted ray lies on the boundary of the surface i.e. angle of refraction is 90∘. Also the sine of the critical angle gives the inverse of the refractive index of the material. We also know that the refractive index μis the ratio of the speed of light in medium v to speed in a vacuumc.
sinC=μ1=cv
Given that critical angle for glass Cg=41∘48′=41.8∘ and that for water Cw=48∘36′=48.6∘.
Then, the refractive index of glassμg=sinCg1=sin(41.8)1=0.6661=1.50
Similarly, the refractive index of waterμw=sinCw1=sin(48.6)1=0.7501=1.33
Then, from the definition of the refractive index, the refractive index of glass-water interface is μgw=μwμg=1.331.50=1.22
The critical of the of glass-water interface is given as sinCgw=μgw1
Cgw=sin−1(μgw1)=sin−1(1.121)=sin−1(0.886)=62.45∘=62∘28′
Hence the answer is A.62∘28′
Note:
To calculate the critical angle we must take the sin−1 of the reflective index and not sinθ1=cosecθ, which are both different. Also note that the refractive index of glass-water interface is μgw=μwμg. The angle of incidence is called the critical angle Cif the angle of the refracted ray lies on the boundary of the surface i.e. angle of refraction is 90∘.