light wavelength 5000 A incident normally on a single .The light by a convex on a sreen placed on focal minimum will be formed the angle of diffraction equal to

Options

A. `0^(0)`
B. `15^(0)`
C. `30^(0)`
D. `50^(0)`

Answer

To solve the problem step by step, we will follow the principles of wave optics, specifically focusing on the diffraction of light through a single slit.

### Step 1: Understand the given data
- Wavelength of light (\( \lambda \)) = 5000 Å (Angstroms)
- Width of the slit (\( d \)) = 0.001 mm

### Step 2: Convert the units
1. Convert the wavelength from Angstroms to meters:
\[ \lambda = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-7} \, \text{m} \]

2. Convert the width of the slit from mm to meters:
\[ d = 0.001 \, \text{mm} = 0.001 \times 10^{-3} \, \text{m} = 1 \times 10^{-6} \, \text{m} \]

### Step 3: Apply the diffraction formula
For a single slit, the condition for the first minimum in the diffraction pattern is given by:
\[ d \sin \theta = m \lambda \]
where \( m \) is the order of the minimum. For the first minimum, \( m = 1 \).

Thus, we can write:
\[ d \sin \theta = \lambda \]

### Step 4: Rearrange the formula to find \( \sin \theta \)
\[ \sin \theta = \frac{\lambda}{d} \]

### Step 5: Substitute the values
Substituting the values of \( \lambda \) and \( d \):
\[ \sin \theta = \frac{5 \times 10^{-7}}{1 \times 10^{-6}} = 0.5 \]

### Step 6: Calculate \( \theta \)
To find \( \theta \), we take the inverse sine:
\[ \theta = \sin^{-1}(0.5) \]

### Step 7: Determine the angle
From trigonometric tables or a calculator, we find:
\[ \theta = 30^\circ \]

### Final Answer
The angle of diffraction for the first minimum is \( 30^\circ \).

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