The refractive index has no unit because it is defined as a ratio of two speeds.
Understanding Refractive Index
Refractive index, often denoted by n, is a fundamental optical property of a material. Because the refractive index of a medium represents the ratio of the speed of light in a vacuum of light speed of the same light within the medium, and ratios do not have any unit. Specifically, it's calculated using the formula:
$n = \frac{c}{v}$
Where:
- c is the speed of light in a vacuum (approximately 299,792,458 meters per second).
- v is the speed of light in the specific medium.
Why Ratios Are Unitless
Quantities that are ratios of two values with the same unit inherently have no unit. Let's look at the formula:
$n = \frac{c}{v}$
The speed of light in a vacuum (c) is measured in units like meters per second (m/s). The speed of light in a medium (v) is also measured in the same units, meters per second (m/s).
When you divide one speed by another, the units cancel out:
$n = \frac{\text{Unit of Speed}}{\text{Unit of Speed}} = \frac{\text{m/s}}{\text{m/s}}$
Just like dividing any number by itself (e.g., 5/5 = 1), dividing a unit by itself results in a unitless value.
Example
Consider calculating the refractive index of water. The speed of light in a vacuum (c) is about $3 \times 10^8$ m/s. The speed of light in water (v) is about $2.25 \times 10^8$ m/s.
$n_{\text{water}} = \frac{3 \times 10^8 \text{ m/s}}{2.25 \times 10^8 \text{ m/s}} = 1.333$
As you can see, the "m/s" units in the numerator and denominator cancel out, leaving a pure number (1.333) with no associated unit.
Practical Insights
- A vacuum has a refractive index of exactly 1, as the speed of light in a vacuum divided by itself is 1.
- All other media typically have a refractive index greater than 1, because the speed of light is always slower in any material medium than in a vacuum.
- The fact that refractive index is unitless makes it a convenient value for comparing how much different materials slow down light relative to a vacuum.
In summary, the refractive index is a dimensionless quantity derived from the ratio of two speeds, leading to the cancellation of their units.
