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Step-by-Step Solution
Step 1: State the problem clearly
We have a Young’s double-slit experiment performed first in air and then in a medium of refractive index $ \mu $. The position of the 8th bright fringe in the medium coincides with the position of the 5th dark fringe in air. We need to find the value of $ \mu $.
Step 2: Recall key formulas for fringe positions
For bright fringes in a Young’s double-slit experiment (in air), the position of the $ m $-th bright fringe is given by:
$ y_{m,\text{air}} = m \frac{\lambda_{\text{air}} D}{d}
$
For dark fringes, the position of the $ n $-th dark fringe in air is:
$ y_{n,\text{dark, air}}
= \left(n - \frac{1}{2}\right) \frac{\lambda_{\text{air}} D}{d}
$
Where:
$ \lambda_{\text{air}} $ is the wavelength of light in air.
$ D $ is the distance between the slits and the screen.
$ d $ is the separation between the two slits.
When the experiment is performed in a medium of refractive index $ \mu $, the wavelength becomes:
$ \lambda_{m} = \frac{\lambda_{\text{air}}}{\mu} $. Thus, for the $ m $-th bright fringe in the medium, the position is:
$ y_{m,\text{medium}}
= m \frac{\lambda_{m} D}{d}
= m \frac{\lambda_{\text{air}}}{\mu} \frac{D}{d}.
$
Step 3: Apply the condition given in the problem
According to the problem, the 8th bright fringe in the medium lies at the same position as the 5th dark fringe in air. Therefore,
$
8 \frac{\lambda_{\text{air}}}{\mu} \frac{D}{d}
= \left(5 - \frac{1}{2}\right) \frac{\lambda_{\text{air}} D}{d}.
$
Simplifying,
$
8 \frac{\lambda_{\text{air}}}{\mu} = 4.5 \lambda_{\text{air}}.
$
$
\frac{8}{\mu} = 4.5
\quad \Longrightarrow \quad
\mu = \frac{8}{4.5} = 1.78.
$
Step 4: State the final answer
The refractive index of the medium is 1.78.
