How does the angular width of principal maximum in the diffraction pattern vary with the width of slit ?

Options

A. Principal maximum increases on increasing the width of slit
B. Principal maximum decreases on decreasing the width of slit
C. Principal maximum increases on decreasing the width of slit
D. None of these

Answer

To determine how the angular width of the principal maximum in a diffraction pattern varies with the width of the slit, we can follow these steps:

### Step 1: Understand the concept of diffraction
Diffraction is the bending of waves around obstacles and the spreading of waves when they pass through small openings (slits). The diffraction pattern produced by a single slit consists of a central maximum and several minima on either side.

### Step 2: Identify the relationship between slit width and minima
The position of the minima in a single-slit diffraction pattern can be described by the formula:
\[ a \sin \theta = n \lambda \]
where:
- \( a \) = width of the slit
- \( \theta \) = angle at which the minima occur
- \( n \) = order of the minima (n = 1, 2, 3,...)
- \( \lambda \) = wavelength of the light used

### Step 3: Approximate for small angles
For small angles, we can use the approximation:
\[ \sin \theta \approx \theta \]
Thus, the equation becomes:
\[ a \theta = n \lambda \]
From this, we can express \( \theta \) as:
\[ \theta = \frac{n \lambda}{a} \]

### Step 4: Determine the angular width of the principal maximum
The angular width of the principal maximum is defined as the angle between the first minima on either side of the central maximum. Therefore, the angular width \( \alpha \) can be expressed as:
\[ \alpha = 2 \theta \]
For the first minima (n = 1):
\[ \alpha = 2 \left(\frac{\lambda}{a}\right) = \frac{2\lambda}{a} \]

### Step 5: Analyze the relationship between angular width and slit width
From the equation \( \alpha = \frac{2\lambda}{a} \), we can see that:
- As the slit width \( a \) increases, the angular width \( \alpha \) decreases.
- Conversely, as the slit width \( a \) decreases, the angular width \( \alpha \) increases.

### Conclusion
Thus, the angular width of the principal maximum in the diffraction pattern varies inversely with the width of the slit. If the slit width increases, the angular width decreases, and if the slit width decreases, the angular width increases.