Question

The optical path length is equal to ____?

Answer 1

Hint: The optical path length is the actual path length times the refractive index. The actual path length measures the distance covered or travelled by the light with the medium being a vacuum. The optical path length measures the distance covered or travelled by the light with the medium other than vacuum.

Complete answer:
The distance that would have been travelled by the light in the vacuum is called the actual path length, whereas, the optical path length is the product of the actual path length and the refractive index of the medium, that is, the distance that would have been travelled by the light in the medium other than vacuum.
The mathematical representation of the same is given as follows.
The distance travelled by the light in a medium is given as follows.
$$x_m={\upsilon }_m\times t$$….. (1)
Where $${\upsilon }_m$$ is the speed of light in medium and t is the time taken.
The distance travelled by the light in a vacuum is given as follows.
$$x_v={\upsilon }_v\times t$$….. (2)
Where $${\upsilon }_v$$ is the speed of light in a vacuum and t is the time taken.
Now represent the equation (2) in terms of time taken.
$$t={\displaystyle \frac{x_v}{{\upsilon }_v}}$$
Substitute the above equation in equation (1)
$$x_m={\upsilon }_m\times {\displaystyle \frac{x_v}{{\upsilon }_v}}$$
The refractive index is a ratio of speed of light in vacuum by the speed of light in medium. So, we have,
$$x_m=n\times x_v$$
$$\therefore$$ The optical path length is equal to the product of the refractive index of the medium and the actual path length.

Note:
The actual path length is also called the geometric path length. The only difference between the optical path length and the actual path length is that the actual path length is taken into consideration when the light travels through the vacuum and the optical path length is taken into consideration when the light travels through medium other than vacuum.