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

To evaluate the given assertion and reason, let's analyze each statement:

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

First, we need to understand the concept of fringe width in Young's double-slit experiment:

• The fringe width is the distance between two successive bright (or dark) fringes on the screen.
• In Young's double-slit experiment, this is calculated using the formula:

\(\beta = \frac{\lambda D}{d}\)

• Here, \(\lambda\) is the wavelength of light used, \(D\) is the distance between the slits and the screen, and \(d\) is the distance between the two slits.
• The fringe width \(\beta\) is the same for both dark and bright fringes, as it solely depends on \(\lambda\), \(D\), and \(d\), not on the nature of the fringe (dark or bright).

Let's now assess the options based on our understanding:

• Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). - This is the correct answer. Both statements are true, and the reason correctly explains why the fringe width for dark and bright fringes is the same, as it is determined by the formula given.
• Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). - This is incorrect because the reason directly explains the assertion about fringe width being the same for both types of fringes.
• Assertion (A) is true, but Reason (R) is false. - This is incorrect because Reason (R) is true; it gives the correct fringe width formula.
• Both Assertion (A) and Reason (R) are false. - This is incorrect because both statements are actually true.

In conclusion, both the assertion and the reason are true, and the reason aptly justifies the assertion.

Answer 2

Picture the interference pattern on the screen as a repeating sequence: bright, dark, bright, dark, evenly laid out one after another. This picture alone is enough to test both statements.

If you measure the gap from one bright fringe to the very next bright fringe, and separately measure the gap from one dark fringe to the very next dark fringe, both gaps turn out to be the exact same length, because the bright and dark fringes alternate at a perfectly regular interval, there is no stretching or compressing of the pattern as you move from a bright region to a dark one. So the claim that dark and bright fringe widths are equal is true.

Now, the reason gives the actual formula for that regular interval, \( \beta = \dfrac{\lambda D}{d} \). This single number is the "step size" of the whole repeating pattern, since the pattern repeats with one constant step size everywhere on the screen, that same step separates bright fringes from each other and dark fringes from each other equally. The formula being independent of fringe type is exactly the geometric reason the pattern looks evenly spaced in the first place.

Since the formula for \( \beta \) is precisely what fixes the constant, repeating spacing responsible for the assertion's claim, the reason is both true and a valid explanation of 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).