Two plane monochromatic coherent waves produce inter- ference pattern (of alternately bright and dark bands) on a screen. When the angle between the two beams is decreased the fringe width

Options

A. Increases
B. is unaffected
C. Decreases
D. Increases and then decreases

Answer

To solve the problem regarding the effect of decreasing the angle between two coherent waves on the fringe width, we can follow these steps:

### Step 1: Understand the Fringe Width Formula
The fringe width (β) in an interference pattern is given by the formula:
\[ \beta = \frac{2\pi a \sin \theta}{\lambda} \]
where:
- \( \beta \) = fringe width
- \( a \) = distance between the slits (or the source of the waves)
- \( \theta \) = angle between the two beams
- \( \lambda \) = wavelength of the light used

### Step 2: Analyze the Effect of Decreasing the Angle
When the angle \( \theta \) is decreased, we need to analyze how this affects the sine function. The sine of an angle decreases as the angle itself decreases. Thus, \( \sin \theta \) will also decrease.

### Step 3: Substitute the Effect into the Formula
Since \( \sin \theta \) decreases when \( \theta \) decreases, we can conclude that:
\[ \beta = \frac{2\pi a \sin \theta}{\lambda} \text{ will also decrease.} \]

### Step 4: Conclusion
Therefore, when the angle between the two beams is decreased, the fringe width \( \beta \) will decrease.

### Final Answer
As the angle between the two beams is decreased, the fringe width decreases.

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