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, we need to understand the relationship between the wavelength of light, the slit width, and the diffraction pattern produced by a single slit. Here’s a step-by-step solution:

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

### Step 2: Set up the equation for the first minima
For the first minima, we can rewrite the equation as:
\[ a \sin(\theta) = \lambda \]
This implies:
\[ \sin(\theta) = \frac{\lambda}{a} \]

### Step 3: Analyze the change in wavelength
According to the problem, the wavelength \( \lambda \) is increased by 50%. Therefore, the new wavelength \( \lambda' \) can be expressed as:
\[ \lambda' = 1.5 \lambda \]

### Step 4: Set up the equation for the unchanged diffraction pattern
To keep the diffraction pattern unchanged, the angle \( \theta \) must remain the same. Thus, we need to find a new slit width \( a' \) such that:
\[ a' \sin(\theta) = \lambda' \]
Substituting for \( \lambda' \):
\[ a' \sin(\theta) = 1.5 \lambda \]

### Step 5: Relate the new slit width to the original slit width
Since \( \sin(\theta) = \frac{\lambda}{a} \), we can substitute this into the equation:
\[ a' \left(\frac{\lambda}{a}\right) = 1.5 \lambda \]
This simplifies to:
\[ a' = 1.5 a \]

### Step 6: Determine the change in slit width
The change in slit width is given by:
\[ \Delta a = a' - a = 1.5 a - a = 0.5 a \]
This means that the slit width must be increased by 50% of its original width.

### Conclusion
Thus, to keep the diffraction pattern unchanged when the wavelength of light is increased by 50%, the slit width must also be increased by 50%.

### Final Answer
The required change in the slit width is **50%**.

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