A microwave of wavelength "2.0cm" falls normally on a slit of width "4.0cm" .The angular spread of the central maxima of the diffraction pattern obtained on a screen "1.5m" away from the slit,will be :

Options

A. `60^@`
B. `15^@`
C. `30^@`
D. `45^@`

Answer

To solve the problem of finding the angular spread of the central maxima of the diffraction pattern, we will follow these steps:

### Step 1: Identify the Given Values
- Wavelength (\( \lambda \)) = 2.0 cm = \( 2.0 \times 10^{-2} \) m
- Width of the slit (\( a \)) = 4.0 cm = \( 4.0 \times 10^{-2} \) m
- Distance to the screen (\( D \)) = 1.5 m

### Step 2: Use the Diffraction Formula
The formula for single-slit diffraction is given by:
\[ a \sin \theta = n \lambda \]
where:
- \( a \) = width of the slit
- \( \theta \) = angle for the first minimum
- \( n \) = order of the minimum (for the first minimum, \( n = 1 \))

### Step 3: Substitute the Values
For the first minimum (\( n = 1 \)):
\[ a \sin \theta = \lambda \]
Substituting the values:
\[ 4.0 \times 10^{-2} \sin \theta = 2.0 \times 10^{-2} \]

### Step 4: Solve for \( \sin \theta \)
Rearranging the equation gives:
\[ \sin \theta = \frac{2.0 \times 10^{-2}}{4.0 \times 10^{-2}} = \frac{1}{2} \]

### Step 5: Find \( \theta \)
Now, we find \( \theta \):
\[ \sin \theta = \frac{1}{2} \implies \theta = 30^\circ \]

### Step 6: Calculate the Angular Spread
The angular spread of the central maxima is given by:
\[ \text{Angular Spread} = 2\theta \]
Substituting the value of \( \theta \):
\[ \text{Angular Spread} = 2 \times 30^\circ = 60^\circ \]

### Final Answer
The angular spread of the central maxima is \( 60^\circ \).

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