A SLIT WHOSE WIDTH IS 5.0cm IS IRRADIATED WITH MICROWAVE OF WAVE LENGTH 2cm.THE ANGULAR SPREAD OF THE CENTRALMAXIMUM IF THE INCIDENCE IS AT AN ANGLE OF 30 DEGREE WITH THE NORMALTO THE PLANE OF THE SLIT.

To determine the angular spread of the central maximum in a single-slit diffraction pattern when microwaves pass through a slit, we can use the principles of wave optics. The angular spread can be calculated using the formula for the first minimum in a single-slit diffraction pattern, which is given by:

Understanding the Formula

The position of the first minimum in the diffraction pattern can be found using the equation:

• a sin(θ) = mλ

Where:

• a = width of the slit
• θ = angle of the first minimum
• m = order of the minimum (for the first minimum, m = 1)
• λ = wavelength of the incident wave

Given Values

In this scenario, we have:

• Width of the slit, a = 5.0 cm = 0.05 m
• Wavelength of the microwave, λ = 2 cm = 0.02 m
• Angle of incidence, 30 degrees

Calculating the Angular Spread

First, we need to find the angle θ for the first minimum. Rearranging the formula gives us:

• sin(θ) = mλ / a

Substituting the values for the first minimum (m = 1):

• sin(θ) = (1 * 0.02 m) / 0.05 m = 0.4

Now, we can find θ:

• θ = sin^-1(0.4)

Calculating this gives:

• θ ≈ 23.58 degrees

Adjusting for the Angle of Incidence

Since the wave is incident at an angle of 30 degrees, we need to adjust our angle θ. The effective angle for the diffraction pattern will be:

• θ' = θ + 30 degrees

Thus:

• θ' ≈ 23.58 degrees + 30 degrees = 53.58 degrees

Final Thoughts on Angular Spread

The angular spread of the central maximum, considering the angle of incidence, is approximately 53.58 degrees. This means that the central maximum will spread out over this angle, which is significant in applications involving wave interference and diffraction.

In practical terms, this spread can affect how we design systems that utilize microwaves, such as antennas or communication devices, where understanding the diffraction pattern is crucial for optimizing performance.