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Question #43736

Difficulty: 8/10 | 🟢 Free

The Question

A single slit of width $a$ is illuminated by a light source. The source emits light with a uniform intensity spectrum over a narrow wavelength range from $\lambda_0 - \frac{\Delta\lambda}{2}$ to $\lambda_0 + \frac{\Delta\lambda}{2}$ , where $\Delta\lambda \ll \lambda_0$ . The total intensity at the center of the central diffraction maximum is $I_0$ . Different wavelengths emitted by the source are mutually incoherent. Which of the following best represents the minimum total intensity of the diffraction pattern near the angular position of the first diffraction minimum of the central wavelength $\lambda_0$ ?

Options

A.
$\frac{I_0}{12} \left( \frac{\Delta\lambda}{\lambda_0} \right)^2$
B.
$0$
C.
$\frac{I_0}{4} \left( \frac{\Delta\lambda}{\lambda_0} \right)^2$
D.
$\frac{I_0}{6} \left( \frac{\Delta\lambda}{\lambda_0} \right)^2$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Single Slit Diffraction Minima|Superposition of Non-identical Waves|Vector Integration

Why This Question Is Tricky

Common Mistakes

Students often mechanically apply the standard single-slit diffraction formula $I = I_0 \text{sinc}^2(\beta)$ and assume that the first minimum always has exactly zero intensity.

This question tests the fundamental superposition principle for non-identical (incoherent) waves.

When a source has a spectral width, different wavelengths produce diffraction patterns with slightly different spatial scales.

Because these wavelengths are mutually incoherent, their intensities (not amplitudes) must be integrated.

The trap is assuming a perfect null exists ( $I=0$ ) or incorrectly evaluating the boundary wavelengths instead of performing the convolution integral, failing to recognize that the overlapping minima of different wavelengths 'blur' the zero-intensity point into a non-zero minimum.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Single Slit Diffraction Minima
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free