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 $ Peff $ will be given respectively as:
(K is a constant of proportionality)
A. $\dfrac{\lambda }{{16}},{P_{eff}} = K{(\dfrac{1}{\lambda })^3} \\ $
B. $\lambda ,{P_{eff}} = K{(\dfrac{1}{\lambda })^2} \\ $
C. $\dfrac{\lambda }{8},{P_{eff}} = K(\dfrac{1}{\lambda }) \\ $
D. $\dfrac{\lambda }{5},{P_{eff}} = K{(\dfrac{1}{\lambda })^{\dfrac{1}{2}}} \\ $

Answer 1

Hint:
To solve this question, we will start with calculating length l of the antenna, as it will be comparable to $\lambda $, only. Then we will find the power radiated by the antenna, we know that, the power that is radiated by linear antenna is inversely proportional to the square of wavelength and directly proportional to the length of antenna, so on putting the value, we will get the required answer.

Complete step by step solution:
We have been given that a signal is to be transmitted through a wave of wavelength $\lambda ,$ using a linear antenna. We need to find the length ‘l’ of the antenna and effective power radiated ‘Peff’.
We know that, Power $ \propto {(\dfrac{l}{\lambda })^2}$
where, \[l{\text{ }} = \] length of the antenna
$\lambda = $ wavelength of wave

For the Peff, i.e., the for the power to be maximum, there is a condition where length of antenna should be greater than at least $\dfrac{\lambda }{4}.$
And if we check the given options.
Option A. $\dfrac{\lambda }{{16}},{P_{eff}} = K{(\dfrac{1}{\lambda })^3}$ where, length of antenna is not greater than $\dfrac{\lambda }{4}.$ So, this option is incorrect.

Option B. $\lambda ,{P_{eff}} = K{(\dfrac{1}{\lambda })^2}$ where, length of antenna is greater than $\dfrac{\lambda }{4}.$ So, this option is correct.

Option C. $\dfrac{\lambda }{8},{P_{eff}} = K(\dfrac{1}{\lambda })$ where, length of antenna is not greater than $\dfrac{\lambda }{4}.$ So, this option is incorrect.

Option D. $\dfrac{\lambda }{5},{P_{eff}} = K{(\dfrac{1}{\lambda })^{\dfrac{1}{2}}}$ where, length of antenna is not greater than $\dfrac{\lambda }{4}.$ So, this option is incorrect.

Therefore, the length l of the antenna and effective power radiated Peff will be given respectively as $\lambda ,{P_{eff}} = K{(\dfrac{1}{\lambda })^2}$.
where, K \[ = \] proportionality constant

Thus, option (B) $\lambda ,{P_{eff}} = K{(\dfrac{1}{\lambda })^2}$, is correct.

Note: In the question, we have been given linear antennas, let us gather some knowledge about them, so, linear antennas are those antennas that use electrically very thin conductors, i.e., the wavelength is much larger than the conductor diameter. And in order to calculate the radiated fields in these antennas, i.e., linear antennas the conductors are modelled as if there is no diameter with the current lines.