29. Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): If Young’s double slit experiment is performed in an optically denser medium than air, then the consecutive fringes come closer. Reason (R): 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 given below: (1) Both (A) and (R) are true and (R) is the correct explanation of (A) (2) (A) is false but (R) is true (3) Both (A) and (R) are true but (R) is not the correct explanation of (A) (4) (A) is true but (R) is false

1. Formula for fringe spacing

In Young’s Double Slit Experiment, the distance between two successive bright (or dark) fringes on the screen is called the fringe width $\beta$.

where

• $\lambda$ = wavelength of light in the medium between the slits and the screen,
• $D$ = distance from slits to screen,
• $d$ = distance between the two slits.

2. What changes in a denser medium?

A medium with refractive index $n$ (>1 for anything denser than air) affects the light wave this way:

• Speed of light: $v = \frac{c}{n}$ (slower than in air).
• Frequency $f$: remains unchanged (source decides it, not the medium).
• Wavelength: $$\lambda' = \frac{v}{f} = \frac{c}{nf} = \frac{\lambda}{n}$$ So the wavelength becomes smaller by a factor $n$.

3. Consequence for fringe width

Since $\beta = \frac{\lambda D}{d}$, inserting $\lambda' = \frac{\lambda}{n}$ gives

Thus $\beta'$ is smaller. The bright/dark bands squeeze closer—supporting the assertion.

4. Linking the reason

Reducing speed ($v$) ➔ reduces wavelength ($\lambda$) (because $f$ does not change) ➔ reduces $\beta$. Therefore, the reason explains the assertion.

What is the question about?

Imagine you have two tiny holes (slits) and you shine light through them onto a screen. The light waves overlap and make bright and dark stripes called fringes. The question asks: “If we do this in a thicker, denser material than air (like water or glass), will those stripes move closer together, and why?”

Key idea for a 10-year-old

1. Light has a wavelength (distance from one wave peak to the next).
2. In thicker stuff (like water), each wave peak is squeezed closer because light slows down.
3. When peaks are closer, the stripes (fringes) on the screen also bunch up.

So, yes, the stripes come closer together because light slows down in a thicker material, even though its “beat speed” (frequency) stays the same.

Step-by-Step Solution

1. Identify relevant formula: $\beta = \frac{\lambda D}{d}$
2. Effect of denser medium:
   • Refractive index $n>1$ ⇒ speed $v = \frac{c}{n}$.
   • Frequency $f$ remains same.
   • Wavelength changes to $\lambda' = \frac{\lambda}{n}$.
3. New fringe width: $\beta' = \frac{\lambda'}{d}D = \frac{\lambda}{n}\frac{D}{d} = \frac{\beta}{n} \; < \; \beta$ Hence fringes come closer. Assertion (A) is true.
4. Check the reason: Statement: "The speed of light reduces in an optically denser medium than air while its frequency does not change."
   • Speed indeed decreases ($v = c/n$).
   • Frequency is invariant. So Reason (R) is true.
5. Does (R) explain (A)? Reduction in speed (with constant frequency) leads to reduced wavelength, thereby reducing $\beta$. Thus (R) correctly explains (A).

Correct option: (1) Both (A) and (R) are true and (R) is the correct explanation of (A).