Q.Derive the laws of refraction from Huygens' principle.

Refractive Index Calculation

The refractive index $n$ of a medium measures how much it slows down and bends light compared to vacuum. This concept collects the standard ways to calculate $n$ from measurable quantities. There is no single formula — you pick the one matching the data you are given.

1. From the speed of light

The defining relation: refractive index is how many times slower light travels in the medium than in vacuum.

where $c = 3\times10^8\ \text{m/s}$ is the speed of light in vacuum and $v$ is its speed in the medium. Since $v \le c$, we always have $n \ge 1$.

Example: light travels at $2\times10^8\ \text{m/s}$ in glass, so $n = (3\times10^8)/(2\times10^8) = 1.5$.

2. From Snell's law (angles of incidence and refraction)

When light goes from medium 1 into medium 2,

For light entering a medium from air ($n_1 \approx 1$):

Measure the angle of incidence $i$ and angle of refraction $r$, take the ratio of their sines.

3. From real and apparent depth

An object under water looks shallower than it is. The refractive index of the liquid is

Example: a coin at the bottom of a tank of real depth $12\ \text{cm}$ appears to be at $9\ \text{cm}$, so $n = 12/9 = 1.33$.

4. From wavelength

Because frequency is unchanged across a boundary while wavelength scales as $1/n$,

5. Relative refractive index

The refractive index of medium 2 with respect to medium 1 is

Note it can be less than 1 (e.g. going from glass to air).

6. From the critical angle

For total internal reflection at a denser-to-rarer boundary with critical angle $C$,

(for the denser medium relative to the rarer one).

Worked example (combining methods)

A ray enters a glass block from air at $i = 45^\circ$ and refracts to $r = 28^\circ$. …