Question

Assertion : In Young’s double-slit experiment, the fringe width for dark and bright fringes is the same. Reason (R): Fringe width is given by \( \beta = \frac{\lambda D}{d} \), where symbols have their usual meanings.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Both Assertion (A) and Reason (R) are false.

Hint:

In YDSE:

Bright and dark fringes are equally spaced.
Fringe width depends only on \( \lambda, D, d \).

Answer 1

The Correct Option is A

Concept: In Young’s double-slit experiment (YDSE):

Fringe width: \[ \beta = \frac{\lambda D}{d} \]
Bright and dark fringes are equally spaced.

Step 1: Analyze Assertion (A). In YDSE:

Distance between two consecutive bright fringes = \( \beta \)
Distance between two consecutive dark fringes = \( \beta \)
Hence, fringe widths are equal. Assertion is true.
Step 2: Analyze Reason (R). The formula: \[ \beta = \frac{\lambda D}{d} \] gives the constant spacing between successive fringes. This applies equally to bright and dark fringes. Reason is true.
Step 3: Relation between A and R. Since the formula directly shows uniform spacing, it explains why bright and dark fringe widths are the same. Conclusion: Both Assertion and Reason are true, and Reason correctly explains Assertion.

Answer 2

This assertion-reason pair can be checked by deriving the fringe-width formula's implication directly rather than just quoting it.

Checking the Assertion: "The fringe width for dark and bright fringes is the same." In Young's double-slit setup, bright fringes occur wherever the path difference between the two slits equals a whole number of wavelengths, and dark fringes occur wherever it equals a half-integer number of wavelengths. Since consecutive bright fringes are spaced by one full wavelength of path difference, and consecutive dark fringes are also spaced by exactly one full wavelength of path difference, both sets of fringes end up spaced identically on the screen. So the assertion is true.

Checking the Reason: "Fringe width is given by \( \beta = \dfrac{\lambda D}{d} \)." This formula for the spacing between consecutive fringes on the screen follows from the geometry of the two-slit setup, and it is a single formula that doesn't distinguish between bright and dark fringes at all, it gives the spacing that applies uniformly to both. So the reason is also true.

Does the Reason explain the Assertion? Since \( \beta \) is one fixed number depending only on \( \lambda \), \( D \), and \( d \), and this same \( \beta \) is what separates both consecutive bright fringes and consecutive dark fringes, the formula itself is precisely why both types of fringes must share the same spacing. So the reason directly and correctly explains the assertion.

Therefore, the correct answer is both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of the Assertion (A).