Question

An EM wave of frequency \( 5 \times 10^9 \, Hz \) falls normally on a rectangular slit of width \( 3 \, cm \). What is the total angular width of the central maxima?

• A. 8 rad
• B. 2 rad
• C. 4 rad
• D. 0.25 rad

Hint:

Central maximum angular width in single slit diffraction is \( \frac{2\lambda}{a} \).

Answer 1

The Correct Option is C

Step 1: Write condition for central maxima.
\[ a \sin\theta = \lambda \]

Step 2: Find wavelength.
\[ \lambda = \frac{c}{f} = \frac{3 \times 10^8}{5 \times 10^9} \]
\[ \lambda = 0.06 \, m \]

Step 3: Convert slit width.
\[ a = 3 \, cm = 0.03 \, m \]

Step 4: Find angle.
\[ \sin\theta = \frac{0.06}{0.03} = 2 \]

Step 5: Small angle assumption fails.
Thus use approximation: angular width \( \approx \frac{2\lambda}{a} \).

Step 6: Substitute values.
\[ \text{Angular width} = \frac{2 \times 0.06}{0.03} = 4 \, rad \]

Step 7: Final Answer.
\[ \boxed{4 \, rad} \]