A glass plate whose index of refraction is 1.67 is immersed in a liquid. The surface of the glass is inclined at an angle of 29.0° with the vertical. A horizontal ray in the glass is incident on the interface. When the liquid is a certain alcohol, the incident ray arrives at the interface at the critical angle. What is the index of refraction of the alcohol?
To find the index of refraction of the alcohol, we can use the concept of the critical angle. The critical angle is the angle of incidence that results in the angle of refraction being 90 degrees (or parallel to the surface).
Given:
Index of refraction of glass (ng) = 1.67
Angle of incidence (θg) = 29.0°
Using Snell's law, we can relate the angles of incidence and refraction to the respective indices of refraction:
ng*sin(θg) = na*sin(θa)
Where:
ng is the index of refraction of the glass
θg is the angle of incidence in the glass
na is the index of refraction of the alcohol
θa is the angle of refraction in the alcohol
At the critical angle, the angle of refraction is 90 degrees:
θa = 90°
Substituting these values into Snell's law:
ng*sin(θg) = na*sin(90°)
Since sin(90°) = 1, the equation simplifies to:
ng*sin(θg) = na
Now we can substitute the known values:
1.67*sin(29.0°) = na
Calculating this expression:
na ≈ 0.807
Therefore, the approximate index of refraction of the alcohol is 0.807.