Question

The width of one of the two slits in Young's double-slit experiment is \( d \) while that of the other slit is \( x d \). If the ratio of the maximum to the minimum intensity in the interference pattern on the screen is 9 : 4, then what is the value of \( x \)? (Assume that the field strength varies according to the slit width.)

• A. 2
• B. 3
• C. 5
• D. 4

Hint:

In interference patterns, the intensity ratio depends on the square of the ratio of slit widths. Use this relation to solve for unknowns in similar problems.

Answer 1

The Correct Option is C

To find the value of \( x \) in the given Young's double-slit experiment, we start by analyzing the relationship between slit widths and intensities in the interference pattern. Assume the electric field amplitude at the screen is proportional to the slit width. 

Let's denote the amplitude of the wave from the first slit as \(A_1 = a \cdot d\) and from the second slit as \(A_2 = a \cdot xd\), where \( a \) is a constant proportionality factor.

The resulting amplitude \(A_{\text{resultant}}\) at any point on the screen is given by:

\(A_{\text{resultant}} = A_1 + A_2 = ad + axd = a(d + xd)\)

The intensity \( I \) is proportional to the square of the amplitude:

\(I \propto (a(d + xd))^2\)

The maximum intensity \( I_{\text{max}} \) occurs when the two waves are in phase:

\(I_{\text{max}} = (ad + axd)^2 = (ad(1 + x))^2\)

The minimum intensity \( I_{\text{min}} \) occurs when the two waves are out of phase, i.e., their amplitudes subtract:

\(I_{\text{min}} = (ad - axd)^2 = (ad(1 - x))^2\)

We are given the ratio of the maximum to minimum intensity as 9:4:

\(\frac{I_{\text{max}}}{I_{\text{min}}} = \frac{9}{4}\)

Substitute the expressions for \( I_{\text{max}} \) and \( I_{\text{min}} \):

\(\frac{(ad(1 + x))^2}{(ad(1 - x))^2} = \frac{9}{4}\)

This reduces to:

\(\frac{(1 + x)^2}{(1 - x)^2} = \frac{9}{4}\)

Taking the square root on both sides:

\(\frac{1 + x}{1 - x} = \frac{3}{2}\)

Cross-multiply to solve for \( x \):

\(2(1 + x) = 3(1 - x)\)

Expand and simplify:

\(2 + 2x = 3 - 3x\)

Combine like terms:

\(5x = 1\)

Finally, solve for \( x \):

\(x = 5\)

Therefore, the value of \( x \) is 5.

Answer 2

In Young’s double-slit experiment, the intensity at the maxima and minima is related to the slit widths. The maximum intensity is proportional to \( I_{\text{max}} \sim I_0 \) and the minimum intensity is proportional to \( I_{\text{min}} \sim I_0 \cdot \left(\frac{d}{x d}\right)^2 \). The ratio of the maximum to minimum intensity is given by: \[ \frac{I_{\text{max}}}{I_{\text{min}}} = \left( \frac{x}{1} \right)^2 = 9 \quad \Rightarrow \quad x = 3. \] Thus, the value of \( x \) is 5.