# How far do electrons go in an antenna?

Do you know what is the distance electrons cover in an antenna during the production of a radio wave?

Does the extension of the oscillation vary with the frequency or the power of the radiation? Is there a relation with amplitude?

If, as I suspect, the width of the oscillation does not exceed 1 cm, can you explain why the same (any) radiowave cannot be effectively produced by a short antenna? Why a good antenna must be (half) the length of the wave, if the electrons cannot possibly oscillate from one end to the other?

I will answer the last question first because I get an unexpected answer to the first question.
An antenna can be thought of as having an impedance - a resistive component and a reactive component.
Power is transmitted to an antenna via a transmission cable which will have its own characteristic impedance often $50$ or $75 \Omega$ from the transmitter which in turn will have a characteristic output impedance..

For maximum power transfer from the transmission line to the antenna there must be an impedance match.
If that does not happen then there will be a reflection and power will be sent back to the transmitter which is undesirable.

The best scenario is to have the impedance of the antenna purely resistive and equal to the impedance of the transmission line.

The impedance of an antenna depends on many factors including the length of the antenna, the geometry of the antenna, the diameter of the conductors which make up the antenna etc.

The graph below shows how the impedance of an antenna varies with its length.

If the length of the dipole is very small compared with the wavelength of the radio wave then the impedance of the antenna (low resistance and very high capacitive reactance) will be a total mismatch for the transmitter. transmitting cable and so the antenna would be a very inefficient radiator.

The following graph is an expanded portion of the graph above.

If the dipole had a length of exactly half a wavelength then its resistance would be quite close to that of transmission cables but there would be some inductive reactance.
The solution is to reduce the length of the dipole until its impedance is purely resistive.
So half wave dipoles are always shorter than half a wavelength.

For the amplitude of motion of an electron I have used the equation $I = n e v A$ where $I$ is the current, $n$ the charge carrier density, $e$ the charge on the charge carrier (electron), $v$ the speed of the charge carrier and $A$ the area of the conductor neglecting the skin effect.

Putting some numbers in $v = \dfrac{I}{neA} = \dfrac {1}{8.48 \times 10^{-28}\times 1.6 \times 10 ^{-19} \times 10^{-5}}\approx 7 \times 10^{-6} \rm m\,s^{-1}$ where the maximum current is $1 \rm A$ passing through a copper conductor of cross-sectional area $10^{-5} \,m^2$

Assuming the motion to be simple harmonic then the maximum displacement is
$\dfrac{v}{2 \pi f} = \dfrac{7 \times 10^{-6}}{2 \pi 10^6} \approx 10^{-12} \rm \, m$ for a $1\,\rm MHz$ signal.
This seems rather small to me.

• @lambertwhite I think that you are asking about bandwidth so look up the following antennas to see that by careful design one change the bandwidth (and directivity) of an antenna: dipole, Yagi, log-periodic etc. – Farcher Mar 8 '17 at 7:06