29. Given below are two statements. Assertion: If Young's double slit experiment is performed in an optically denser medium than air, then the consecutive fringes come closer. Reason: The speed of light reduces in an optically denser medium than air while its frequency does not change. In the light of the above statements, choose the most appropriate answer from the options below. Option 1: Both Assertion and Reason are true and Reason is the correct explanation of Assertion. Option 2: Assertion is false but Reason is true. Option 3: Both Assertion and Reason are true but Reason is not the correct explanation of Assertion. Option 4: Assertion is true but Reason is false.

Key Concepts Needed

1. Fringe Width ($\beta$)
   In Young’s Double-Slit Experiment, the separation between adjacent bright (or dark) fringes on the screen is
   $\beta = \frac{\lambda D}{d}$
   where
   • $\lambda$ = wavelength of light in the medium
   • $D$ = distance between double slit and screen
   • $d$ = separation between the two slits

2. Light in a Denser Medium
   • The refractive index of the medium is $\mu$.
   • Speed becomes $v = \frac{c}{\mu}$ (slower than in air).
   • Frequency $f$ does not change when light enters a new medium.
   • Using $v = f \lambda$, the wavelength becomes
   $\lambda_{\text{medium}} = \frac{v}{f} = \frac{c}{\mu f} = \frac{\lambda_{\text{air}}}{\mu}$

3. Effect on Fringe Width
   Substituting $\lambda_{\text{medium}}$ into the fringe-width formula,
   $\beta_{\text{medium}} = \frac{\lambda_{\text{air}} / \mu}{d} D = \frac{\beta_{\text{air}}}{\mu}$
   Since $\mu > 1$ (optically denser than air), $\beta_{\text{medium}}$ is smaller → fringes crowd together.

Linking Assertion & Reason

• Assertion: Fringes come closer (smaller $\beta$) in a denser medium → True.
• Reason: Light slows down but frequency stays the same in a denser medium → True.
• The Reason directly explains why wavelength (hence fringe width) shrinks.
Therefore, both statements are true and the Reason correctly explains the Assertion.

🎨 Imagine Painting Stripes on Water!

Suppose you drop two pebbles in a calm pool. They make ripples that overlap and produce bright rings (crests) and dark rings (troughs). Those rings are like the bright and dark fringes we see on a screen in Young’s Double-Slit Experiment.

Now, if the pool suddenly becomes thick like honey (optically denser), the ripples travel slower. Because the speed drops but the number of ripples produced each second (frequency) stays the same, the distance between successive ripples shrinks. On the wall (our screen) the bright and dark stripes come closer together.

So, saying “fringes come closer in a denser medium” is true, and explaining it by “light slows down while its frequency stays unchanged” is also true. That slowing-down directly causes the fringes to squeeze together.

Step-by-Step Working

1. Fringe Width in Air $\beta_{\text{air}} = \frac{\lambda_{\text{air}} D}{d}$

2. Fringe Width in Denser Medium ($\mu$)
   Wavelength becomes $\displaystyle \lambda_{\text{medium}} = \frac{\lambda_{\text{air}}}{\mu}$. $\beta_{\text{medium}} = \frac{\lambda_{\text{medium}} D}{d} = \frac{\lambda_{\text{air}} D}{\mu d} = \frac{\beta_{\text{air}}}{\mu}$ Because $\mu > 1$, $\beta_{\text{medium}} < \beta_{\text{air}}$ → fringes are closer.

3. Truth Table • Assertion – True.
   • Reason – True (speed ↓, frequency same).
   • Reason explains Assertion.

4. Correct Option: Option 1 — Both Assertion and Reason are true and Reason is the correct explanation of Assertion.