Question

In Young's double slit experiment, if the distance between the slits is halved and the distance to the screen is doubled, then the fringe width becomes
• Halved
• Doubled
• Four times
• Remain the same

• A. Halved
• B. Doubled
• C. Four times
• D. Remain the same

Hint:

In Young's double slit experiment: \[ \beta=\frac{\lambda D}{d} \] Remember:
• Increasing \(D\) increases fringe width
• Increasing \(d\) decreases fringe width

Answer 1

The Correct Option is C

Concept: In Young's Double Slit Experiment (YDSE), the fringe width is given by: :contentReference[oaicite:3]{index=3} where:
• \(\beta\) = fringe width
• \(\lambda\) = wavelength of light
• \(D\) = distance between slit and screen
• \(d\) = distance between slits From the formula:
• Fringe width is directly proportional to \(D\)
• Fringe width is inversely proportional to \(d\)

Step 1: Write the original fringe width expression. Initially: \[ \beta=\frac{\lambda D}{d} \] Suppose:
• Original slit separation \(=d\)
• Original screen distance \(=D\)

Step 2: Apply the changes given in the question. According to the question:
• Distance between slits is halved \[ d'=\frac{d}{2} \]
• Distance to screen is doubled \[ D'=2D \]

Step 3: Substitute the new values into fringe width formula. New fringe width: \[ \beta'=\frac{\lambda D'}{d'} \] Substituting: \[ \beta'=\frac{\lambda(2D)}{d/2} \] Now simplify carefully. Dividing by \(d/2\) means multiplying by \(2/d\): \[ \beta'=\lambda(2D)\times\frac{2}{d} \] \[ \beta'=\frac{4\lambda D}{d} \] But: \[ \beta=\frac{\lambda D}{d} \] Therefore: \[ \boxed{ \beta'=4\beta } \]

Step 4: Interpret the result physically. The fringe width becomes: \[ \boxed{ 4 \text{ times the original fringe width} } \] This happens because:
• Increasing screen distance spreads fringes apart.
• Decreasing slit separation also spreads fringes apart. Both effects increase fringe width simultaneously.

Step 5: Choose the correct option. Hence the correct answer is: \[ \boxed{ \text{Four times} } \] which corresponds to: \[ \boxed{(3)} \] Additional Understanding: In YDSE: \[ \beta \propto \frac{D}{d} \] Therefore:
• Larger \(D\) gives wider fringes.
• Smaller \(d\) gives wider fringes. Final Conclusion: The fringe width becomes: \[ \boxed{ 4 \text{ times} } \] Hence, the correct answer is: \[ \boxed{(3)} \]