(Radiation resistance / Radiated power / Radiation resistance force)
In this
section, Feynman discusses the radiation resistance and radiated power of an
antenna, as well as the force of radiation resistance that is also known as Abraham–Lorentz force.
1. Radiation
resistance:
“In
fact, if it is well built it will appear as almost a pure resistance, with
very little inductance or capacitance, because we would like to radiate as
much energy as possible out of the antenna. This resistance that an antenna
shows is called the radiation resistance (Feynman et al., 1963, p. 32–1).”
In the Audio Recordings* [3 min: 10 sec] of this
lecture, Feynman says: “In fact, if it is well built it will appear as almost a pure resistance
because we would like to radiate as much energy as possible out of the antenna.”
That is, he did not explicitly mention “with very little inductance or
capacitance,” but this is added in the edited Feynman’s lecture. On the contrary, electrical engineers may define an antenna as an inductor that
contains resistance (ohmic loss) or parasitic capacitance as well as explain
the importance of impedance in the antenna. Furthermore, the antenna’s resonant
frequency occurs when its capacitance and inductance cancel each other. In other words, the antenna will appear as
almost a pure resistance because the net reactance (inductance + capacitance) becomes almost
zero at the resonant frequency.
*The Feynman Lectures Audio
Collection: https://www.feynmanlectures.caltech.edu/flptapes.html
At the beginning of the lecture, Feynman mentions
that “Today’s lecture is on two different subjects, one is called radiation
damping or radiation resistance, the other is the scattering of light.” Perhaps
he could have defined radiation resistance as Rrad = 2P/Io2, where Io is the peak instantaneous
current and it is the same everywhere on the antenna. However, the term “radiation reaction” is almost always concerned with the
oscillation of only an electron that cannot be exactly determined by
experiments. Simply put, it is an oversimplification to model the
phenomenon of radiation reaction using only an electron. This is an
idealization because it is also difficult to compute the ohmic losses and
dielectric losses of the antenna. Thus, it should be better to study the interactions as well as collisions among
electrons and copper ions within the antenna in a broader sense or with
better experimental sensibility.
2. Radiated power:
“The rate at which power is radiated
by the antenna is proportional to the square of the current in the antenna,
of course, because all the fields are proportional to the currents, and the
energy liberated is proportional to the square of the field. The coefficient of
proportionality between radiated power and <I2> is
the radiation resistance (Feynman et al., 1963, p. 32–1).”
Feynman explains that the radiated power of the
antenna is proportional to the square of the alternating current because
all the fields are proportional to the currents, and the energy liberated is
proportional to the square of the field. However, it may not be clear why the fields
are directly proportional to the currents. For example, one may add that the net
electric field is also proportional to the number of charges (i.e., copper ions
or electrons). Specifically, the total electric field is partly proportional to
1/r, 1/r^2, and 1/r^3 depending on the distance (r)
from the antenna. More important, the radiated power P = <I2>R could be related to the formula S = ϵ0c<E2> that was derived in the last chapter using the theory of refractive
index and low-density approximation. (The formula S = ϵ0c<E2> is also known as a form of Poynting vector.)
When
we say “We can calculate—” that is not quite right—we cannot, because we
have not yet studied the laws of electricity at short distances; only at large
distances do we know what the electric field is. We saw the formula (28.3), but at present it is too complicated for us to calculate the
fields inside the wave zone (Feynman et al., 1963, p. 32–1).”
The term wave zone is not commonly used,
however, Feynman defines it as “the region beyond a few wavelengths (Feynman et
al., 1963, p. 29-3).” Alternatively, one may use the term near-field region,
intermediate-field region, and far-field region. In general, the far-field decays in
accordance with 1/r, whereas near-field decays faster as 1/r^3. By
using S = ϵ0c<E2>, we can show that the antenna’s radiated power
through a spherical area (4pr^2) would approach zero for large r,
e.g., no observable radiation for near-field that varies with 1/r^3. On
the other hand, the radiated power is measurable for far-field that varies with
1/r.
3. Radiation
resistance force:
“This
model of the origin of the resistance to acceleration, the radiation resistance
of a moving charge, has run into many difficulties, because our present view of
the electron is that it is not a “little ball”; this problem has never
been solved. Nevertheless we can calculate exactly, of course, what the net radiation
resistance force must be, i.e., how much loss there must be when we
accelerate a charge, in spite of not knowing directly the mechanism of how that
force works (Feynman et al., 1963, p. 32–2).”
According to Feynman, the origin of the radiation resistance force on a
moving charge is an unsolved problem because the electron is not a
“little ball.” Historically, the radiation resistance force is known as Abraham–Lorentz force, or
the self-force of an electron. Strictly speaking, our present view of the
electron is not definitely a “little ball” because its size may vary
depending on the experiment. In Feynman Lectures on gravitation, he remarked that “[t]here is evidently some trouble here, since we have
inherited a prejudice that an accelerating charge should radiate, whereas we do
not expect a charge lying in a gravitational field to radiate (Feynman et al.,
1995, p. 123).” Importantly, he clarifies that the self-force formula derived from Poynting’s theorem is only
valid for cyclic motions (instead of constant acceleration).
In a sense, one may need a
little pride to explain the radiation resistance force by developing a
field-less theory of electromagnetism. In his Nobel
Lecture, Feynman reveals: “…I suggested to myself, that
electrons cannot act on themselves, they can only act on other electrons. That
means there is no field at all……there is not an infinite number of degrees of
freedom in the field. There is no field at all; or if you insist on thinking in
terms of ideas like that of a field, this field is always completely determined
by the action of the particles which produce it……And, like falling in love with
a woman, it is only possible if you do not know much about her, so you cannot
see her faults. The faults will become apparent later, but after the love is
strong enough to hold you to her. So, I was held to this theory, in spite of all
difficulties, by my youthful enthusiasm.”
Review Questions:
1. How would you define the radiation resistance of
an antenna (Does the antenna have inductance and capacitance)?
2.
How would you explain all the fields are
proportional to the current in the
antenna?
3. How would you explain the force of radiation
resistance?
The moral of the lesson: The radiated power of an
antenna is depending on its radiation resistance whereby its capacitance and
inductance cancel each other at the resonant
frequency.
References:
1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading,
MA: Addison-Wesley.
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