Certainly! To find the index of refraction for cubic zirconia, we will use the fundamental relationship between the speed of light in a vacuum and the speed of light in the material. The index of refraction [tex]\( n \)[/tex] is defined as:
[tex]\[ n = \frac{c}{v} \][/tex]
where:
- [tex]\( c \)[/tex] is the speed of light in a vacuum, approximately [tex]\( 3.00 \times 10^8 \)[/tex] meters per second (m/s).
- [tex]\( v \)[/tex] is the speed of light in the material, which is given as [tex]\( 1.36 \times 10^8 \)[/tex] meters per second (m/s) for cubic zirconia.
Let's proceed step-by-step:
1. Identify the speed of light in a vacuum ([tex]\( c \)[/tex]):
[tex]\[ c = 3.00 \times 10^8 \, \text{m/s} \][/tex]
2. Identify the speed of light in cubic zirconia ([tex]\( v \)[/tex]):
[tex]\[ v = 1.36 \times 10^8 \, \text{m/s} \][/tex]
3. Use the formula for the index of refraction:
[tex]\[ n = \frac{c}{v} \][/tex]
4. Substitute the given values into the formula:
[tex]\[ n = \frac{3.00 \times 10^8}{1.36 \times 10^8} \][/tex]
5. Calculate the numerical value:
[tex]\[ n = \frac{3.00}{1.36} \approx 2.2058823529411766 \][/tex]
Therefore, the index of refraction for cubic zirconia, given that light travels through it at a speed of [tex]\( 1.36 \times 10^8 \,\text{m/s} \)[/tex], is approximately [tex]\( 2.21 \)[/tex].
