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

Key Concepts to Know

1. Fringe Width ($\beta$)

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

   where $\lambda$ is the wavelength of light in the medium, $D$ is the distance from the slits to the screen, and $d$ is the slit separation.

2. Light in a Different Medium

   • Speed: $v = \frac{c}{n}$ ( $n$ = refractive index )
   • Frequency: unchanged when light enters a new medium.
   • Wavelength: $\lambda = \frac{v}{f} \;\Rightarrow\; \lambda = \frac{c}{n f} = \frac{\lambda_0}{n}$.
     Hence $\lambda$ becomes smaller by a factor $n$.

3. Connecting to Fringe Width
   Because $\beta \propto \lambda$, a smaller $\lambda$ in a denser medium leads to a smaller fringe width → fringes come closer.

Logical Chain a Student Should Follow

• Recognize that fringe spacing depends on $\lambda$.
• Recall that entering a denser medium lowers speed but frequency stays the same.
• Deduce that $\lambda$ therefore decreases.
• Conclude that smaller $\lambda$ ⇒ smaller $\beta$ ⇒ closer fringes.
• Notice that the Reason statement (speed ↓, frequency same) is exactly the fact used to reach the Assertion, making it a correct explanation.

What’s happening here?

Imagine you are throwing pebbles in a pond from two spots side-by-side. The waves they create bump into each other and make a pattern of bright and dark rings on the water. In light, this happens as bright and dark fringes on a screen.

Now pretend the water suddenly becomes thicker like honey. The waves slow down, so the distance between the rings becomes smaller.

In the light experiment, when we put everything into a denser material (like glass or water instead of air), light slows down but its beat (frequency) stays the same. Because of that, each light wave becomes shorter and the bright–dark stripes move closer together. That is exactly what the assertion is saying.

Step-by-Step Solution

1. Fringe Width in YDSE

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

2. Effect of a Denser Medium
   • Speed becomes $v = c/n$
   • Frequency $f$ remains unchanged
   • Wavelength becomes $\lambda' = \frac{v}{f} = \frac{c}{nf} = \frac{\lambda_0}{n}$
   where $\lambda_0$ is the wavelength in air.

3. New Fringe Width

   $$\beta' = \frac{\lambda' D}{d} = \frac{\lambda_0 D}{n d} = \frac{\beta_0}{n}$$

   Because $n > 1$, we have $\beta' < \beta_0$. Consecutive fringes are closer together. Thus Assertion (A) is true.

4. Checking the Reason
   The statement that the speed of light reduces while frequency stays constant in a denser medium is correct and it directly leads to the decrease in wavelength that explains (A). Therefore Reason (R) is true and the correct explanation.

5. Choosing the Option
   Option (1): Both (A) and (R) are true and (R) is the correct explanation of (A).
   is the right choice.

Final Answer: Option (1)