Question

A slit of width \(a\) is illuminated by light of wavelength \(\lambda\). The linear separation between 1st and 3rd minima in the diffraction pattern produced on a screen placed at a distance \(D\) from the slit system is ______.

• A. \(\frac{D\lambda}{a}\)
• B. \(1.5 \frac{D\lambda}{a}\)
• C. \(\frac{2D\lambda}{a}\)
• D. \(\frac{3D\lambda}{a}\)

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In single-slit diffraction, minima occur at specific angles depending on the slit width and the wavelength of light. The linear position on the screen can be found using the small-angle approximation.
Step 2: Key Formula or Approach:
The condition for the \(n^{\text{th}}\) minimum in single-slit diffraction is:
\(a \sin \theta_n = n\lambda\) \quad where \(n = 1, 2, 3, \dots\)
For small angles, \(\sin \theta_n \approx \tan \theta_n = \frac{y_n}{D}\), where \(y_n\) is the linear distance from the central maximum on the screen.
Therefore, the position of the \(n^{\text{th}}\) minimum is given by:
\(y_n = \frac{n \lambda D}{a}\).
Step 3: Detailed Explanation:
For the 1st minimum (\(n = 1\)):
\(y_1 = \frac{(1) \lambda D}{a} = \frac{\lambda D}{a}\).
For the 3rd minimum (\(n = 3\)):
\(y_3 = \frac{(3) \lambda D}{a} = \frac{3\lambda D}{a}\).
The linear separation between the 1st and 3rd minima is the difference between their respective positions:
\(\Delta y = y_3 - y_1 = \frac{3\lambda D}{a} - \frac{\lambda D}{a} = \frac{2\lambda D}{a}\).
Step 4: Final Answer:
The linear separation is \(2 \frac{D\lambda}{a}\).