Why is an antenna designed to be one-half of the wavelength of the wave it is supposed to receive?
In my book all it says is that the voltage is larger when the antenna is 1/2 the wavelength of the wave.
It is explained in detail here
http://farside.ph.utexas.edu/teaching/jk1/lectures/node82.html
The important point is that an antenna that is much shorter than the wavelength will have a radiation resistance that will be far less than the internal radiation.
Then for every joule of energy transmitted in the form of radio-waves, you'll necessarily have dissipated many more joules of energy in the form of heat.
If an antenna is of the order of a wavelength, then the radiation resistance will be quite large. To use an antenna for transmission, the impedance of the antenna needs to be the same as that of the transmitter (to be precise, it has to be the complex conjugate), because otherwise the output of the transmitter will be partally reflected back into the transmitter, potentially damaging the transmitter.
Now, since the impedance of a half wavelength antenna is of the same order as that of typical transmitters and coax cables, this makes them easy to use.
But in practice, an antenna tuner will be used to match the impedance of the antenna to that of the transmitter, so it is not necessary to have an antenna that is exactly the "right length".
Typo in first sentece:
I meant to say:
The important point is that an antenna that is much shorter than the wavelength will have a radiation resistance that will be far less than the internal ohmic resistance.
The length of an antenna being one-half of the wavelength it is supposed to receive is based on a principle called resonance. Resonance occurs when the natural frequency of an antenna matches the frequency of the incoming wave. When the antenna is resonant, it efficiently captures the energy from the wave and converts it into electrical signals.
To understand why an antenna is designed to be one-half of the wavelength, let's start with the basics. An antenna is essentially a conductive structure that radiates or receives electromagnetic waves. When an electromagnetic wave passes through an antenna, it induces an alternating current (AC) in the antenna.
For a given frequency of the wave, the current induced in the antenna depends on the length or size of the antenna. At certain lengths, the AC current produced by the wave reaches its maximum value. This happens when the antenna's length is equal to one-half of the wavelength of the incoming wave.
At this resonant length, the voltage across the antenna terminals is also the largest. The voltage is directly proportional to the current, so when the current is at its maximum, the voltage will also be maximum. This is why your book mentions that the voltage is larger when the antenna is one-half the wavelength.
By designing the antenna to be one-half of the wavelength, we ensure that it is resonant and can efficiently capture the energy from the incoming wave. This enables the antenna to receive and transmit signals effectively.
It's worth noting that a half-wavelength antenna is not the only type of antenna used in practice. There are various antenna designs for different purposes, and their lengths may deviate from this ideal half-wavelength requirement. However, for general applications, a half-wavelength antenna is a commonly used and efficient choice.