If the wavelength of light diffracted by a single slit is increased by 50%, what change in the slit width will leave the diffraction pattern unchanged?

Answer

To solve the problem of how the slit width must change when the wavelength of light diffracted by a single slit is increased by 50%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship**: The condition for the first minimum in a single slit diffraction pattern is given by the formula:
\[ a \sin \theta = m \lambda \]
where:
- \( a \) is the width of the slit,
- \( \theta \) is the angle of diffraction,
- \( m \) is the order of the minimum (for the first minimum, \( m = 1 \)),
- \( \lambda \) is the wavelength of light.

2. **Initial conditions**: Let's denote the initial wavelength as \( \lambda \) and the initial slit width as \( a \). The equation for the first minimum becomes:
\[ a \sin \theta = \lambda \]

3. **Change in wavelength**: If the wavelength is increased by 50%, the new wavelength \( \lambda' \) can be expressed as:
\[ \lambda' = 1.5 \lambda \]

4. **New condition for unchanged diffraction pattern**: We want the diffraction pattern to remain unchanged, which means that \( \theta \) must remain the same. Thus, the new equation for the first minimum with the new wavelength becomes:
\[ a' \sin \theta = \lambda' \]
where \( a' \) is the new width of the slit.

5. **Substituting the new wavelength**: Substituting \( \lambda' \) into the equation gives:
\[ a' \sin \theta = 1.5 \lambda \]

6. **Relating the two conditions**: Since \( a \sin \theta = \lambda \), we can replace \( \lambda \) in the new equation:
\[ a' \sin \theta = 1.5 (a \sin \theta) \]

7. **Canceling out \( \sin \theta \)**: Assuming \( \sin \theta \neq 0 \), we can divide both sides by \( \sin \theta \):
\[ a' = 1.5 a \]

8. **Conclusion**: This means that the new slit width \( a' \) must be 1.5 times the original slit width \( a \). Therefore, if the wavelength of light is increased by 50%, the slit width must also increase by 50% to keep the diffraction pattern unchanged.

### Final Answer:
The slit width must be increased by 50%.

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