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: Understanding the Concept:
In a double-slit diffraction pattern, the overall intensity is a product of the interference pattern from two point sources and the diffraction pattern from a single slit. A "missing order" occurs when an interference maximum is supposed to appear at the same angle as a diffraction minimum. At this angle, the intensity is zero due to the diffraction minimum, so the interference fringe "disappears" or is missing.
Step 2: Key Formula or Approach:
Let \(a\) be the slit width and \(d\) be the distance 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 be satisfied for the same angle \(\theta\).
Dividing the two equations gives 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: Detailed Explanation:
We are given:
Slit width, \( a = 0.12 \) mm.
Distance between slits, \( d = 0.6 \) mm.
First, calculate the ratio \(d/a\):
\[ \frac{d}{a} = \frac{0.6 \, \text{mm}}{0.12 \, \text{mm}} = 5 \] The condition for missing orders becomes:
\[ \frac{n}{m} = 5 \implies n = 5m \] Now, we find the values of the missing orders \(n\) by substituting integer values for \(m\) (where \( m = 1, 2, 3, \ldots \)):

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 missing orders are 5, 10, 15, 20, ...
There seems to be a typo in the question or the options, as none of the options match this result. However, if we assume there was a typo in the question and the slit separation `d` was meant to be 0.72 mm, let's see the result:
\[ \frac{d}{a} = \frac{0.72 \, \text{mm}}{0.12 \, \text{mm}} = 6 \] Then the condition for missing orders would be:
\[ n = 6m \] The 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) matches option (A). Given the options, it is highly probable that this was the intended question.
Step 4: Final Answer:
Assuming a typo in the problem statement where \(d=0.72\) mm instead of \(d=0.6\) mm, the missing orders are 6, 12, 18, 24.