Understanding the Concept:
In a Young's Double Slit Experiment setup, the linear width of the interference fringes ($\beta$) is given by:
\[
\beta = \frac{\lambda D}{d} \implies \beta \propto \lambda
\]
When the entire experimental physical apparatus is submerged in a liquid medium of refractive index $\mu$, the physical speed of light drops, causing its operational wavelength to shorten to $\lambda' = \frac{\lambda}{\mu}$. Consequently, the fringe width decreases by the same index factor:
\[
\beta' = \frac{\beta}{\mu}
\]
Step 1: Substitute values directly into the scaling expression.
We are given:
Initial fringe width, $\beta = 1.82\text{ mm}$
Refractive index of the liquid, $\mu = 1.3$
Let's compute the modified fringe width ($\beta'$):
\[
\beta' = \frac{1.82\text{ mm}}{1.3} = 1.4\text{ mm}
\]