Question

A signal is to be transmitted through a wave of wavelength \(\lambda\), using a linear antenna. The length l of the antenna and effective power radiated \(P_{eff}\) will be given respectively as :
(K is a constant of proportionality)

• A. \(\lambda,\ P_{eff} = K\left(\frac{1}{\lambda}\right)^{2}\)
• B. \(\frac{\lambda}{8},\ P_{eff} = K\left(\frac{1}{\lambda}\right)\)
• C. \(\frac{\lambda}{16},\ P_{eff} = K\left(\frac{1}{\lambda}\right)^{3}\)
• D. \(\frac{\lambda}{5},\ P_{eff} = K\left(\frac{1}{\lambda}\right)^{\frac{1}{2}}\)

Hint:

An antenna radiates efficiently only if its length is comparable to the signal wavelength, and the radiated power varies as \((l/\lambda)^2\), i.e. as \(1/\lambda^2\) for a given length.

Answer 1

The Correct Option is A

For effective transmission the length of the antenna must be of the order of the wavelength, \(l \sim \lambda\) (a length much smaller than \(\lambda\), such as \(\lambda/16\), radiates very poorly).
The power radiated by an antenna of length l is proportional to \((l/\lambda)^2\); for \(l\) of the order of \(\lambda\) this gives \(P_{eff}\propto\left(\frac{1}{\lambda}\right)^2\), so \(P_{eff} = K\left(\frac{1}{\lambda}\right)^2\).
Only option (1) has both \(l = \lambda\) and the inverse-square dependence.
Answer: (1) \(\lambda,\ P_{eff} = K\left(\frac{1}{\lambda}\right)^{2}\)