Angular width `(beta)` of central maximum of a diffraction pattern on a single slit does not depend upon

Options

A. Distance between slit and source
B. Wavelength of the light used
C. Width of the slit
D. Frequency of liht used

Answer

To solve the question regarding the angular width (β) of the central maximum of a diffraction pattern on a single slit, we need to analyze the factors that influence this width based on the principles of wave optics.

### Step-by-Step Solution:

1. **Understanding the Diffraction Pattern**:
The diffraction pattern produced by a single slit is characterized by a central maximum and several minima on either side. The angular width of the central maximum is defined as the angle between the first minima on either side of the central maximum.

2. **Using the Diffraction Condition**:
The condition for minima in single-slit diffraction is given by:
\[ d \sin \theta = m \lambda \]
where:
- \(d\) = width of the slit
- \(\lambda\) = wavelength of the light
- \(m\) = order of the minimum (for the first minimum, \(m = 1\))

3. **Finding the Angle**:
For the first minimum, we can express this as:
\[ \sin \theta = \frac{\lambda}{d} \]
For small angles, we can approximate \(\sin \theta \approx \theta\) (in radians), leading to:
\[ \theta \approx \frac{\lambda}{d} \]

4. **Calculating the Angular Width**:
The angular width (β) of the central maximum is twice the angle to the first minimum:
\[ \beta = 2\theta = 2 \left(\frac{\lambda}{d}\right) = \frac{2\lambda}{d} \]

5. **Identifying Dependencies**:
From the equation \(\beta = \frac{2\lambda}{d}\), we can see that:
- The angular width β **depends on** the wavelength \(\lambda\) (directly proportional).
- The angular width β **depends on** the slit width \(d\) (inversely proportional).

6. **Determining What It Does Not Depend On**:
The angular width β does not depend on the distance between the slit and the observation screen (denoted as \(D\)). This is because the angular width is determined solely by the slit width and the wavelength of the light used.

### Conclusion:
Thus, the angular width (β) of the central maximum of a diffraction pattern on a single slit does not depend on the distance between the slit and the source (or screen).

### Final Answer:
The angular width (β) of the central maximum does not depend on the distance between the slit and the source/screen.

---