Physical Background
The electrical field created by an electric charge is
![\vec E={-q\over 4\pi \varepsilon_\circ}\left[{\vec e_{r'}\over r'^2}+
{r'\over c}{d\ \over dt}\left({\vec e_{r'}\over r'^2}\right) +
{1\over c^2}{d^2\ \over dt^2}\left(\vec e_{r'}\right)\right]\,](http://upload.wikimedia.org/math/f/2/7/f27b9f10cec8557fb4b79a472c57ac25.png)
where:
- is the speed of light in vacuum.
- is the permittivity of free space.
- is the distance from the observation point (the place where is evaluated) to the point where the charge was seconds before the time when the measure is done.
- is the unit vector directed from the observation point (the place where is evaluated) to the point where the charge was seconds before the time when the measure is done.
The "prime" in this formula appears because the electromagnetic signal travels at the speed of light. Signals are observed as coming from the point where they were emitted and not from the point where the emitter is at the time of observation. The stars that we see in the sky are no longer where we see them. We will see their current position years in the future; some of the stars that we see today no longer exist.
The first term in the formula is just the electrostatic field with retarded time.
The second term is as though nature were trying to allow for the fact that the effect is retarded (Feynman).
The third term is the only term that accounts for the far field of antennas.
The two first terms are proportional to . Only the third is proportional to .
Near the antenna, all the terms are important. However, if the distance is large enough, the first two terms become negligible and only the third remains:
If the charge q is in sinusoidal motion with amplitude and pulsation the power radiated by the charge is:
-
- watts.
Note that the radiated power is proportional to the fourth power of the frequency. It is far easier to radiate at high frequencies than at low frequencies. If the motion of charges is due to currents, it can be shown that the (small) electrical field radiated by a small length of a conductor carrying a time varying current is
The left side of this equation is the electrical field of the electromagnetic wave radiated by a small length of conductor. The index reminds that the field is perpendicular to the line to the source. The reminds that this is the field observed seconds after the evaluation on the current derivative. The angle is the angle between the direction of the current and the direction to the point where the field is measured.
The electrical field and the radiated power are maximal in the plane perpendicular to the current element. They are zero in the direction of the current.
Only time-varying currents radiate electromagnetic power.
If the current is sinusoidal, it can be written in complex form, in the same way used for impedances. Only the real part is physically meaningful:
where:
- is the amplitude of the current.
- is the angular frequency.
The (small) electric field of the electromagnetic wave radiated by an element of current is:
And for the time :
The electric field of the electromagnetic wave radiated by an antenna formed by wires is the sum of all the electric fields radiated by all the small elements of current. This addition is complicated by the fact that the direction and phase of each of the electric fields are, in general, different.
Read more about this topic: Antenna Measurement
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