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 determine the value of \( x \) in the provided Young's double-slit experiment, the relationship between slit widths and interference pattern intensities is examined. It is assumed that the electric field amplitude at the screen is directly proportional to the slit width.

The amplitude of the wave from the first slit is defined as \(A_1 = a \cdot d\), and from the second slit as \(A_2 = a \cdot xd\), where \( a \) represents a constant proportionality factor.

The resultant amplitude \(A_{\text{resultant}}\) at any point on the screen is calculated as:

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

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

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

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

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

Minimum intensity \( I_{\text{min}} \) occurs when the waves are out of phase, resulting in amplitude subtraction:

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

The ratio of maximum to minimum intensity is given as 9:4:

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

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

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

This simplifies to:

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

Taking the square root of both sides yields:

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

Cross-multiplication is performed to solve for \( x \):

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

Expansion and simplification results in:

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

Combining like terms gives:

\(5x = 1\)

Solving for \( x \) yields the final value:

\(x = 5\)

The value of \( x \) is determined to be 5.