Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 22 - Wave Optics - Exercises and Problems - Page 651: 60

Answer

a) $915\;\rm m$ b) $0.152\;\rm W/m^2$

Work Step by Step

$$\color{blue}{\bf [a]}$$ The width of the central maximum, in a circular aperture, is given by $$w=\dfrac{2.44\lambda L}{D}$$ This width is the diameter of the radar beam at a distance of $L$. Recalling that $c=\lambda f$, hence $\lambda=c/f$; $$w=\dfrac{2.44c L}{fD}$$ Plugging the known; $$w=\dfrac{2.44(3\times 10^8)(30\times 10^3)}{(12\times 10^9)(2)}$$ $$w=\color{red}{\bf 915}\;\rm m$$ $$\color{blue}{\bf [b]}$$ The average intensity is given by $$I=\dfrac{P}{A}$$ where $A$ is the area and $P$ is the power. Hence, $$I=\dfrac{P}{\pi r^2}$$ Plugging the known; $$I=\dfrac{100\times 10^3}{\pi \left(\frac{915}{2}\right)^2}$$ $$I=\color{red}{\bf 0.152}\;\rm W/m^2$$
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