A parallel beam of light of wavelength `4000 Å` passes through a slit of width `5 xx 10^(-3) m`. The angular spread of the central maxima in the diffraction pattern is

Options

A. `1.6 xx 10^(-3) rad`
B. `1.6 xx 10^(-4) rad`
C. `3.2 xx 10^(-3) rad`
D. `3.2 xx 10^(-4) rad`

Answer

To solve the problem of finding the angular spread of the central maxima in the diffraction pattern when a parallel beam of light passes through a slit, we will follow these steps:

### Step 1: Convert Wavelength to Meters
The given wavelength of light is \( 4000 \) Å (angstroms). We need to convert this to meters.

\[ \text{Wavelength} (\lambda) = 4000 \, \text{Å} = 4000 \times 10^{-10} \, \text{m} = 4 \times 10^{-7} \, \text{m} \]

### Step 2: Identify the Slit Width
The width of the slit is given as:

\[ a = 5 \times 10^{-3} \, \text{m} \]

### Step 3: Use the Formula for Angular Width
The half angular width of the central maximum in a single-slit diffraction pattern can be approximated using the formula:

\[ \sin \theta \approx \theta \approx \frac{\lambda}{a} \]

### Step 4: Substitute Values into the Formula
Now, substituting the values of \( \lambda \) and \( a \):

\[ \theta \approx \frac{4 \times 10^{-7}}{5 \times 10^{-3}} \]

### Step 5: Calculate the Half Angular Width
Calculating the above expression:

\[ \theta \approx \frac{4}{5} \times 10^{-4} \, \text{radians} \]

### Step 6: Calculate the Total Angular Width
Since \( \theta \) represents half of the angular width, the total angular width of the central maximum is:

\[ \text{Total Angular Width} = 2\theta = 2 \times \left(\frac{4}{5} \times 10^{-4}\right) = \frac{8}{5} \times 10^{-4} \, \text{radians} \]

### Step 7: Final Calculation
Calculating the total angular width gives:

\[ \text{Total Angular Width} = 1.6 \times 10^{-4} \, \text{radians} \]

### Conclusion
Thus, the angular spread of the central maxima in the diffraction pattern is:

\[ \boxed{1.6 \times 10^{-4} \, \text{radians}} \]