Index of Refraction

Question

The scenario:

A ray of light strikes the center of the (a) flat surface and (b) curved surface of a semicircular glass medium with the angle of incidences in degrees of 10, 20, 30, 40 and 50.

The angles of refraction for each angle of incidence and for each surface were:

(a) flat surface

Angle of incidence (deg) | Angle of Refraction (deg) | Index of refraction
10                       | 6                         | 1.66
20                       | 12                        | 1.65
30                       | 19                        | 1.54
40                       | 25                        | 1.52
50                       | 30                        | 1.53

(b) curved surface

Angle of incidence (deg) | Angle of Refraction (deg) 
10                       | 16  
20                       | 31 
30                       | 50 
40                       | 73  
50                       | N/A 

I used the formula: index of refraction $=\sin(\theta_{incident}) / \sin(\theta_{refracted})$

Questions:

a). Do I still use the same formula to solve for the index of refraction for the curved surface? Because if I use it, I'll get a different index of refraction.

b). If ever they are really different, why are the calculated indexes of refraction for both media are different for each angle of incidence? Is the difference expected?

Thank you!

Answer 1

Snell's law still applies to the curved surface, but you have to measure the angles of incidence and refraction relative to the surface where the light hits.

The image is my attempt to show parallel rays of light falling on a curved surface. Even though the rays are parallel, the angle of incidence is different for the two rays because it has to be measured relative to the normal at the point the light strikes the surface. Hence the angle $i$ is not the same as the angle $i'$.

Response to comment:

It has become clear from the comments that the problem is that the value of $n$ depends on whether the light is passing from the air to glass or from glass to air. To be precise the two refractive indices are reciprocals of each other i.e.

$$ n_{air-glass} = \frac{1}{n_{glass-air}} $$

The refraction of the light ray happens because the speed of light, and therefore the wavelength, changes when the light enters and leaves the glass. The refractive index when a light ray passes from a medium 1 to a medium 2 is:

$$ n_{1-2} = \frac{v_1}{v_2} $$

where $v_1$ is the speed of light in medium 1 and $v_2$ is the speed of light in medium 2. So in our example the refractive index when passing from medium 2 to medium 1 is:

$$ n_{2-1} = \frac{v_2}{v_1} $$

i.e.

$$ n_{1-2} = \frac{1}{n_{2-1}} $$