Question

Which of the following orders in a double slit Fraunhofer diffraction pattern will be missing if the slit width is 0.12 mm and slits are 0.6 mm apart?

• A. 6, 12, 18, 24
• B. 4, 8, 12, 16
• C. 3, 4, 5, 6
• D. 3, 11, 15

Hint:

For missing order problems, the key is the ratio \(d/a\). If \(d/a = k\), then the missing interference orders are \(n = k, 2k, 3k, \ldots\). Always calculate this ratio first. If your answer doesn't match the options, re-check for possible typos in the given values that would lead to one of the options.

Answer 1

The Correct Option is A

Step 1: Concept Clarification:
A double-slit diffraction pattern results from the combined effects of interference between two point sources and diffraction from a single slit. A "missing order" occurs when an interference maximum coincides with a diffraction minimum. At this point, the intensity becomes zero due to the diffraction minimum, causing the interference fringe to disappear.
Step 2: Governing Equations:
Let \(a\) represent the slit width and \(d\) represent the separation between the centers of the slits.
The condition for the \(n^{th}\) order interference maximum is:
\[ d \sin\theta = n\lambda, \quad \text{where } n = 0, 1, 2, \ldots \] The condition for the \(m^{th}\) order diffraction minimum is:
\[ a \sin\theta = m\lambda, \quad \text{where } m = 1, 2, 3, \ldots \] For an order \(n\) to be missing, both conditions must hold simultaneously for the same angle \(\theta\).
Dividing the two equations yields the condition for missing orders:
\[ \frac{d \sin\theta}{a \sin\theta} = \frac{n\lambda}{m\lambda} \implies \frac{d}{a} = \frac{n}{m} \] Step 3: Calculation and Analysis:
Given values:
Slit width, \( a = 0.12 \) mm.
Slit separation, \( d = 0.6 \) mm.
First, compute the ratio \(d/a\):
\[ \frac{d}{a} = \frac{0.6 \, \text{mm}}{0.12 \, \text{mm}} = 5 \] The condition for missing orders simplifies to:
\[ \frac{n}{m} = 5 \implies n = 5m \] Substituting integer values for \(m\) (where \( m = 1, 2, 3, \ldots \)) determines the missing orders \(n\):

For \(m=1\), \(n = 5(1) = 5\). The 5th order is missing.

For \(m=2\), \(n = 5(2) = 10\). The 10th order is missing.

For \(m=3\), \(n = 5(3) = 15\). The 15th order is missing.
The sequence of missing orders is 5, 10, 15, 20, ...
A discrepancy is noted between the calculated results and typical multiple-choice options, suggesting a potential typo in the problem statement. If we assume \(d = 0.72\) mm instead of \(d = 0.6\) mm:
\[ \frac{d}{a} = \frac{0.72 \, \text{mm}}{0.12 \, \text{mm}} = 6 \] In this scenario, the condition for missing orders becomes:
\[ n = 6m \] The corresponding missing orders would be:

For \(m=1\), \(n = 6\).

For \(m=2\), \(n = 12\).

For \(m=3\), \(n = 18\).

For \(m=4\), \(n = 24\).
This sequence (6, 12, 18, 24) aligns with option (A). Therefore, it is highly probable that the intended value for \(d\) was 0.72 mm.
Step 4: Conclusion:
Under the assumption of a typo in the original problem statement, with \(d=0.72\) mm instead of \(d=0.6\) mm, the missing orders are identified as 6, 12, 18, 24.