What will be the angle of diffraction for the first secondary maximum due to diffraction at a single slit of width 0.5 mm and using light off 5000 Å?

Options

A. `1.5xx10^(-4)` radian
B. `1.5xx10^(-3)` radian
C. `0.75xx10^(-3)` radian
D. 0.001 radian

Explanation

To find the angle of diffraction for the first secondary maximum due to diffraction at a single slit, we can follow these steps:

### Step 1: Understand the given values
- Width of the slit (a) = 0.5 mm = \(0.5 \times 10^{-3}\) m
- Wavelength of light (\(\lambda\)) = 5000 Å = \(5000 \times 10^{-10}\) m = \(5 \times 10^{-7}\) m

### Step 2: Identify the formula for the angle of diffraction
For a single slit diffraction, the condition for the first secondary maximum is given by the formula:
\[ a \sin \theta = \frac{(2n + 1) \lambda}{2} \]
where \(n\) is the order of the maximum. For the first secondary maximum, \(n = 1\).

### Step 3: Substitute the values into the formula
Substituting \(n = 1\) into the formula:
\[ a \sin \theta = \frac{(2 \cdot 1 + 1) \lambda}{2} = \frac{3 \lambda}{2} \]

### Step 4: Rearrange the equation to solve for \(\sin \theta\)
Now we can write:
\[ \sin \theta = \frac{3 \lambda}{2a} \]

### Step 5: Substitute the values of \(\lambda\) and \(a\)
Substituting the values of \(\lambda\) and \(a\):
\[ \sin \theta = \frac{3 \times (5 \times 10^{-7})}{2 \times (0.5 \times 10^{-3})} \]

### Step 6: Calculate the right side
Calculating the right side:
\[ \sin \theta = \frac{15 \times 10^{-7}}{1 \times 10^{-3}} = 15 \times 10^{-4} \]

### Step 7: Find the angle \(\theta\)
Since \(\sin \theta \approx \theta\) for small angles (in radians), we can write:
\[ \theta \approx 15 \times 10^{-4} \text{ radians} \]

### Step 8: Final answer
Thus, the angle of diffraction for the first secondary maximum is:
\[ \theta = 1.5 \times 10^{-3} \text{ radians} \]