Step 1: Understanding the Concept:
Optical path length is defined as the product of the geometric distance a light ray travels and the refractive index of the medium through which it travels.
When a transparent medium (like a mica film) is placed in the path of a light ray in a vacuum or air, the light travels slower in the medium, effectively increasing the optical path length compared to traveling the same geometric distance in a vacuum.
Step 2: Key Formula or Approach:
The optical path equivalent to a geometric distance $t$ in a medium of refractive index $n$ is $n \cdot t$.
The geometric distance replaced by this medium was $t$, which had an optical path of $1 \cdot t$ (assuming air/vacuum).
The change in optical path $\Delta x$ is:
\[ \Delta x = \text{New Optical Path} - \text{Old Optical Path} \]
Step 3: Detailed Explanation:
Before the film is introduced, the light travels a distance $t$ through the air (refractive index $\approx 1$).
Old optical path for that section $= 1 \cdot t = t$.
After the thin mica film is introduced, the light travels the same geometric distance $t$ through the film (refractive index $n$).
New optical path for that section $= n \cdot t$.
The additional optical path introduced by the film is the difference between the new and old optical paths:
\[ \text{Increase in optical path} = n \cdot t - t \]
Factor out the common term $t$:
\[ \text{Increase} = (n - 1)t \]
Since $n>1$ for any physical medium like mica, the quantity $(n-1)t$ is positive, indicating an increase in the optical path.
Step 4: Final Answer:
The optical path is increased by $(n - 1)t$.