Chapter 36: Q31P (page 1111)

Millimetre-wave radar generates a narrower beam than conventional microwave radar, making it less vulnerable to antiradar missiles than conventional radar. (a) Calculate the angular width $2 θ$ of the central maximum, from first minimum to first minimum, produced by a 220 GHz radar beam emitted by a 55.0-cm diameter circular antenna. (The frequency is chosen to coincide with a low-absorption atmospheric “window.”) (b) What is $2 θ$ for a more conventional circular antenna that has a diameter of 2.3 m and emits at wavelength 1.6 cm?

Expert verified

(a) The angular width $2 θ$ of the central maximum is $0.346 °$ .

(b) The value of $2 θ$ for a more conventional circular antenna is $0.97 °$ .

Step by step solution

The angle corresponding to the first minima in diffraction pattern is given by,

$s i n θ = 1.22 \frac{λ}{d} θ = s i n^{- 1} (1.22 \frac{λ}{d})$ ….. (1)

Find the wavelength as follows.

$λ = \frac{c}{f} = \frac{3 \times 10^{8}}{220 \times 10^{9}} = 1.36 \times 10^{- 3} m$

Substitute $0.55 m$ for d and $1.36 \times 10^{- 3} m$ for $λ$ in equation (1).

$θ = \sin^{- 1} (\frac{1.22 (1.36 \times 10^{- 3} m)}{0.55 m}) θ = \sin^{- 1} (3.0167 \times 10^{- 3}) θ = 0.1728 2 θ \approx 0.346 °$

Therefore, the angular width $2 θ$ of the central maximum is $0.346 °$ .

Substitute $2.3 m$ for d and $1.6 \times 10^{- 2} m$ for $λ$ in equation (1).

$θ = \sin^{- 1} (\frac{1.22 (1.6 \times 10^{- 2} m)}{2.3 m}) θ = \sin^{- 1} (8.487 \times 10^{- 3}) θ = 0.486 ° 2 θ \approx 0.97 °$

Therefore, the value of $2 θ$ for a more conventional circular antenna is $0.97 °$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.