Topic: Electromagnetics
A lossless antenna has directional gain $G(\theta,\varphi)$ then $\int_{\varphi=-\pi/2}^{\varphi=\pi/2}\int_{\theta=-\pi/2}^{\theta=\pi/2}G(\theta,\varphi)d\theta~d\varphi$ is
- A. $4\pi$
- B. $2\pi$
- C. $\le 4\pi$
- D. $\le 2\pi$
Answer: C. $\le 4\pi$
Explanation: The integral of the directional gain over all solid angles (the entire sphere, i.e., $4\pi$ steradians) is equal to $4\pi$. Since the given integration limits ($-\pi/2$ to $\pi/2$) cover only a portion of the entire sphere, the integral evaluates to a value less than or equal to $4\pi$.