A parallel beam of monochromatic light is incident normally on a slit. The diffraction parttern is observed on a screen placed at the focal plane of convex lens. If the slit width is increased , the central maximum of the diffraction pattern will

Options

A. Become broader and fainter
B. Become broader and bright
C. Become narrower and brighter
D. Become narrower and brighter

Answer

To solve the problem step by step, we need to analyze the diffraction pattern created by a single slit and how it changes when the slit width is altered.

### Step 1: Understanding the Diffraction Pattern
When a parallel beam of monochromatic light passes through a single slit, it creates a diffraction pattern on a screen. The central maximum is the brightest part of this pattern, and it is flanked by alternating dark and bright fringes.

### Step 2: Condition for Minima
The condition for the minima in a single slit diffraction pattern is given by the formula:
\[ D \sin \theta = n \lambda \]
where:
- \( D \) is the width of the slit,
- \( \theta \) is the angle at which the minima occur,
- \( n \) is the order of the minima (n = 1, 2, 3,...),
- \( \lambda \) is the wavelength of the light.

### Step 3: Angular Width of the Central Maximum
For small angles (where \( \theta \) is small), we can use the approximation \( \sin \theta \approx \theta \). Thus, the condition for the first minima becomes:
\[ D \theta = \lambda \]
From this, we can express \( \theta \) as:
\[ \theta = \frac{\lambda}{D} \]

### Step 4: Width of the Central Maximum
The width of the central maximum can be defined as the distance between the first minima on either side of the central maximum. Therefore, the total angular width of the central maximum is:
\[ \text{Width} = 2 \theta = 2 \frac{\lambda}{D} \]

### Step 5: Effect of Increasing Slit Width
If the slit width \( D \) is increased, the width of the central maximum will be affected. From the equation \( \text{Width} = 2 \frac{\lambda}{D} \), we can see that as \( D \) increases, the width of the central maximum decreases.

### Conclusion
Thus, if the slit width is increased, the central maximum of the diffraction pattern will become narrower.

### Final Answer
The central maximum will become narrower and brighter.

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