Plane microwaves are incident on a long slit having a width of 5.0 cm. Calculate the wavelength of the microwaves if the first diffraction minimum is formed at `theta=30^@`.

Explanation

To solve the problem, we need to calculate the wavelength of the microwaves given the width of the slit and the angle at which the first diffraction minimum occurs. Here are the steps to find the solution:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Width of the slit (b) = 5.0 cm = 5.0 × 10^-2 m
- Angle for the first diffraction minimum (θ) = 30°

2. **Understand the Formula for Diffraction Minimum:**
- The condition for the first minimum in single-slit diffraction is given by the formula:
\[ b \sin \theta = n \lambda \]
- Here, \( n \) is the order of the minimum. For the first minimum, \( n = 1 \).

3. **Convert the Angle to Radians (if necessary):**
- However, since we will be using the sine function, we can directly use the angle in degrees.

4. **Calculate the Sine of the Angle:**
- Calculate \( \sin 30° \):
\[ \sin 30° = \frac{1}{2} \]

5. **Substitute the Values into the Formula:**
- Substitute \( b \), \( \sin \theta \), and \( n \) into the formula:
\[ 5.0 \times 10^{-2} \sin 30° = 1 \cdot \lambda \]
\[ 5.0 \times 10^{-2} \cdot \frac{1}{2} = \lambda \]

6. **Solve for the Wavelength (λ):**
- Calculate \( \lambda \):
\[ \lambda = \frac{5.0 \times 10^{-2}}{2} = 2.5 \times 10^{-2} \text{ m} \]

7. **Final Answer:**
- The wavelength of the microwaves is:
\[ \lambda = 2.5 \times 10^{-2} \text{ m} \text{ or } 25 \text{ cm} \]