Correct Option: (iii) \(\frac{\beta}{\mu}\)

Step 1 (Formula / Governing Law & Given Data): [0.5 Mark]

The fringe width \(\beta\) in a Young's Double Slit Experiment (YDSE) is given by the formula:

$$ \beta = \frac{\lambda D}{d} $$

Where:

• \(\lambda\) = Wavelength of light in the medium.
• \(D\) = Distance between the slits and the screen.
• \(d\) = Distance between the two slits.

Step 2 (Step-by-Step Mathematical Working): [0.25 Mark]

When the apparatus is immersed in a liquid of refractive index \(\mu\), the wavelength of light changes to \(\lambda'\), given by:

$$ \lambda' = \frac{\lambda}{\mu} $$

The new fringe width \(\beta'\) becomes:

$$ \beta' = \frac{\lambda' D}{d} $$

Substituting the value of \(\lambda'\) into the equation:

$$ \beta' = \frac{(\lambda / \mu) D}{d} = \frac{1}{\mu} \left( \frac{\lambda D}{d} \right) $$

Step 3 (Final Concluded Answer): [0.25 Mark]

Since \(\beta = \frac{\lambda D}{d}\), we substitute this back into the equation:

$$ \beta' = \frac{\beta}{\mu} $$

Thus, the new fringe width is \(\frac{\beta}{\mu}\).

💡 EL10 Concept Insight: Think of light waves like a marching band; when they enter a denser liquid, they slow down and their "steps" (wavelength) get shorter, which naturally makes the interference pattern on the screen shrink by the factor of the liquid's refractive index.