The Index of Refraction

Historical context:
- French physicist Jean Foucault conducted the first measurement of the speed of light in a medium (excluding vacuum or air) in 1862.
- Foucault measured the speed of light in water, finding it to be 2.25 × 10^8 m/s.
- The speed of light varies among different media, each being slower than in a vacuum.

Definition: The index of refraction (n) of a medium is defined as the ratio of the speed of light in a vacuum (c) to the speed of light in that medium (v).
Mathematical expression:
- The formula is expressed as:
  n = \frac{c}{v}
- Where:
  - n = index of refraction
  - c = speed of light in a vacuum
  - v = speed of light in a given medium
Units:
- The units for c and v are both in m/s, resulting in the dimensionless nature of n (units cancel out).

The index of refraction can also be calculated using the sines of angles of light as it transitions between media.
Alternative formula:
- When light travels from a vacuum into a transparent medium, it can be described as:
  n = \frac{\sin(\thetai)}{\sin(\thetaR)}
- Where:
  - \theta_i = angle of incidence in the vacuum
  - \theta_R = angle of refraction in the medium

Table 1 provides the index of refraction values for numerous materials:
- Air/Vacuum: n = 1.00
- Ice: n = 1.31
- Pure Water: n = 1.33
- Ethyl Alcohol: n = 1.36
- Quartz: n = 1.46
- Vegetable Oil: n = 1.47
- Olive Oil: n = 1.48
- Acrylic: n = 1.49
- Glass: n = 1.52
- Zircon: n = 1.92
- Diamond: n = 2.42

Example: Calculate the index of refraction for sodium chloride (salt) given:
- c = 3.00 × 10^8 m/s
- v_{sodium ext{ chloride}} = 1.96 × 10^8 m/s
Required: Calculate n.
Analysis and Solution:
- Use the formula:
  n = \frac{c}{v} = \frac{3.00 × 10^8 m/s}{1.96 × 10^8 m/s}
- Solve for n:
  - n = 1.53
Statement: The index of refraction for sodium chloride (salt) is approximately 1.53.

Example: Calculate the speed of light in olive oil given:
- c = 3.00 × 10^8 m/s
- n_{olive ext{ oil}} = 1.48
Required: Calculate v.
Analysis and Solution:
- Rearrange the formula:
  n = \frac{c}{v}
- Which gives:
  - v = \frac{c}{n}
  - Substitute in values:
  - v = \frac{3.00 × 10^8 m/s}{1.48}
  - v = 2.03 × 10^8 m/s
Statement: The speed of light in olive oil is approximately 2.03 × 10^8 m/s.

The symbol “c” for the speed of light originates from the Latin term “celeritas,” which means velocity.

The index of refraction is defined as:
- The ratio of the speed of light in a vacuum to the speed of light in a medium:
  n = \frac{c}{v}
- Also represented through the angles of incidence and refraction:
  n = \frac{\sin(\thetai)}{\sin(\thetaR)}
The index of refraction is a dimensionless quantity due to the cancellation of units.

(a) Define the term “index of refraction.”
(b) Discuss why it is a dimensionless quantity.
Calculate the index of refraction for vinegar with a speed of light of 2.30 × 10^8 m/s .
Based on a speed of light of 1.69 × 10^8 m/s , find the index of refraction for sapphire.
Use Table 1 to find the speed of light in (a) quartz and (b) diamond.
For an 80% sugar solution with an index of refraction of 1.49, calculate the speed of light in this solution.
For an acetone index of refraction of 1.36, what is the speed of light in acetone?
For a substance with a speed of light of 2.20 × 10^8 m/s , calculate the index of refraction and determine its possible identity using Table 1.
If the calculated speed of light in an unknown substance is 4.00 × 10^8 m/s , analyze how you could identify a calculation error.
When a light ray travels from diamond into air with an angle of refraction of 56°, detail how the angle changes if glass is introduced next to the diamond and justify based on speed changes. Include a ray diagram.
Examine why the index of refraction is considered a unique property of a medium.