Electromagnetic Waves, Antennas, and RF: Radiation from Time-Varying Currents and the Hertzian Dipole
ElectromagneticWavesAntennasAndRFRadiationFromTimeVaryingCurrentsAndTheHertzianDipole
2026-10-10 00:08:25
2026-10-10 00:08:25
Topic
electromagnetic radiation
time-varying current
retarded potential
Hertzian dipole
infinitesimal dipole
current element
near field
reactive field
induction field
far field
radiation field
radiation pattern
Poynting vector
radiation intensity
radiated power
radiation resistance
directivity
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\section*{Electromagnetic Waves, Antennas, and RF: Radiation from Time-Varying Currents and the Hertzian Dipole}
EM22 described electromagnetic energy guided along a transmission line. EM23 asks what happens when a time-varying current distribution is allowed to launch fields into open space. The simplest source that contains the essential physics is the \emph{Hertzian dipole}: an ideal current element whose length is much smaller than a wavelength. Although no physical antenna is literally infinitesimal, this source is the basic building block for thin-wire antennas, radiation integrals, arrays, and aperture methods \cite{Balanis2016,Stutzman2012,Griffiths2017,Jackson1999}.
The central chain is
\begin{equation}
\boxed{
\begin{aligned}
\text{time-varying current}
&\longrightarrow \text{retarded potential}
\longrightarrow \mathbf E,\mathbf H,\\
&\longrightarrow \left(r^{-3},r^{-2},r^{-1}\right)
\longrightarrow \text{radiation pattern and power}.
\end{aligned}
}
\end{equation}
The most important physical distinction is that not every field surrounding an antenna is a radiated field. Close to the source, electric and magnetic energy can be stored and returned to the source each cycle. Far away, the surviving $1/r$ terms carry net power outward.
\section{Time convention and propagation convention}
Throughout this article, sinusoidal quantities use the phasor convention
\begin{equation}
\mathbf E(\mathbf r,t)
=\Re\left\{\mathbf E(\mathbf r)e^{i\omega t}\right\}.
\end{equation}
An outward traveling spherical wave therefore contains
\begin{equation}
e^{-ikr},
\end{equation}
where
\begin{equation}
k=\omega\sqrt{\mu\epsilon}=\frac{2\pi}{\lambda}
\end{equation}
and, in a lossless medium,
\begin{equation}
\eta=\sqrt{\frac{\mu}{\epsilon}}.
\end{equation}
In vacuum,
\begin{equation}
\eta_0\approx376.73\,\Omega,
\qquad
k_0=\frac{\omega}{c}.
\end{equation}
Writing the convention explicitly is important because reversing the sign convention for time also reverses several $i$ signs in the phasor formulas, while leaving all measurable powers unchanged.
\section{Why radiation must be retarded}
Maxwell's equations do not allow a change in source current to influence distant points instantaneously. In a homogeneous medium with propagation speed
\begin{equation}
v=\frac{1}{\sqrt{\mu\epsilon}},
\end{equation}
the field observed at position $\mathbf r$ depends on the source at the earlier \emph{retarded time}
\begin{equation}
\boxed{
t_r=t-\frac{R}{v},
}
\end{equation}
where
\begin{equation}
\mathbf R=\mathbf r-\mathbf r',
\qquad
R=|\mathbf R|.
\end{equation}
The time-domain magnetic vector potential is
\begin{equation}
\boxed{
\mathbf A(\mathbf r,t)
=\frac{\mu}{4\pi}
\int
\frac{\mathbf J(\mathbf r',t-R/v)}{R}
\,dV'.
}
\end{equation}
For sinusoidal steady state, retardation becomes a phase factor. The phasor vector potential is
\begin{equation}
\boxed{
\mathbf A(\mathbf r)
=\frac{\mu}{4\pi}
\int
\mathbf J(\mathbf r')
\frac{e^{-ikR}}{R}
\,dV'.
}
\end{equation}
The factor $1/R$ represents spherical amplitude spreading, while $e^{-ikR}$ records propagation phase. This is the first mathematical appearance of the radiating $1/r$ behavior that will dominate the far field.
\begin{center}
\includegraphics{EM23_fig01_retarded_current_element.png}
\vspace{0.4em}
\textit{Figure. A time-varying current element at the source point influences an observation point only after the propagation delay $R/v$. In sinusoidal steady state, that delay becomes the phase factor $e^{-ikR}$.}
\end{center}
\section{The Hertzian current element}
Place a very short current element of length $\ell$ at the origin and orient it along the $z$ axis. Let its current phasor be $I_0$. The defining assumptions are
\begin{equation}
\ell\ll\lambda
\end{equation}
and
\begin{equation}
r\gg\ell,
\end{equation}
so the current element may be treated as spatially infinitesimal at the observation point.
Its current moment is
\begin{equation}
\boxed{I_0\ell}
\end{equation}
with units of ampere-meters. For an ideal Hertzian element the current is taken uniform over $\ell$.
The associated end charges are not optional. Charge conservation requires
\begin{equation}
\nabla\cdot\mathbf J
+\frac{\partial\rho}{\partial t}=0.
\end{equation}
For harmonic time dependence,
\begin{equation}
\nabla\cdot\mathbf J+i\omega\rho=0.
\end{equation}
Thus a time-varying short current element necessarily has a corresponding oscillating charge distribution. The electric near field will contain the effect of those charges.
For the ideal element, the vector potential is
\begin{equation}
\boxed{
\mathbf A
=\hat{\mathbf z}
\frac{\mu I_0\ell}{4\pi r}
e^{-ikr}.
}
\end{equation}
Define
\begin{equation}
A_0(r)=\frac{\mu I_0\ell}{4\pi r}e^{-ikr}.
\end{equation}
Since
\begin{equation}
\hat{\mathbf z}
=\hat{\mathbf r}\cos\theta
-\hat{\boldsymbol\theta}\sin\theta,
\end{equation}
the spherical components are
\begin{equation}
A_r=A_0\cos\theta,
\qquad
A_\theta=-A_0\sin\theta,
\qquad
A_\phi=0.
\end{equation}
\section{Deriving the magnetic field}
The magnetic flux density is
\begin{equation}
\mathbf B=\nabla\times\mathbf A.
\end{equation}
For this axisymmetric source, only the $\phi$ component survives. The spherical-coordinate curl gives
\begin{equation}
(\nabla\times\mathbf A)_\phi
=\frac{1}{r}
\left[
\frac{\partial(rA_\theta)}{\partial r}
-\frac{\partial A_r}{\partial\theta}
\right].
\end{equation}
First,
\begin{equation}
rA_\theta
=-\frac{\mu I_0\ell}{4\pi}
e^{-ikr}\sin\theta,
\end{equation}
so
\begin{equation}
\frac{\partial(rA_\theta)}{\partial r}
=\frac{i k\mu I_0\ell}{4\pi}
e^{-ikr}\sin\theta.
\end{equation}
Also,
\begin{equation}
\frac{\partial A_r}{\partial\theta}
=-A_0\sin\theta.
\end{equation}
Therefore
\begin{equation}
B_\phi
=\frac{\mu I_0\ell\sin\theta}{4\pi}
\left(
\frac{ik}{r}
+\frac{1}{r^2}
\right)e^{-ikr}.
\end{equation}
Since $\mathbf B=\mu\mathbf H$,
\begin{equation}
\boxed{
H_\phi
=\frac{I_0\ell\sin\theta}{4\pi}
\left(
\frac{ik}{r}
+\frac{1}{r^2}
\right)e^{-ikr}.
}
\end{equation}
Two radial dependences already appear:
\begin{equation}
H_\phi
\sim
\frac{1}{r^2}
+\frac{1}{r}.
\end{equation}
The $1/r^2$ term is associated with the induction field; the $1/r$ term survives into the radiation zone.
\section{Deriving the electric field}
Outside the source region, $\mathbf J=0$, so the phasor Ampere-Maxwell law is
\begin{equation}
\nabla\times\mathbf H
=i\omega\epsilon\mathbf E.
\end{equation}
Hence
\begin{equation}
\boxed{
\mathbf E
=\frac{1}{i\omega\epsilon}
\nabla\times\mathbf H.
}
\end{equation}
Because $H_\phi$ is the only nonzero magnetic-field component,
\begin{equation}
(\nabla\times\mathbf H)_r
=\frac{1}{r\sin\theta}
\frac{\partial}{\partial\theta}
\left(\sin\theta H_\phi\right)
\end{equation}
and
\begin{equation}
(\nabla\times\mathbf H)_\theta
=-\frac{1}{r}
\frac{\partial}{\partial r}(rH_\phi).
\end{equation}
Carrying out the derivatives and using
\begin{equation}
\eta=\sqrt{\frac{\mu}{\epsilon}},
\qquad
k=\omega\sqrt{\mu\epsilon},
\end{equation}
gives
\begin{equation}
\boxed{
E_r
=\frac{\eta I_0\ell\cos\theta}{2\pi}
\left(
\frac{1}{r^2}
-\frac{i}{kr^3}
\right)e^{-ikr},
}
\end{equation}
\begin{equation}
\boxed{
E_\theta
=\frac{\eta I_0\ell\sin\theta}{4\pi}
\left(
\frac{ik}{r}
+\frac{1}{r^2}
-\frac{i}{kr^3}
\right)e^{-ikr},
}
\end{equation}
and
\begin{equation}
\boxed{E_\phi=0.}
\end{equation}
Together with
\begin{equation}
\boxed{
H_\phi
=\frac{I_0\ell\sin\theta}{4\pi}
\left(
\frac{ik}{r}
+\frac{1}{r^2}
\right)e^{-ikr},
}
\end{equation}
these are the complete external fields of the ideal Hertzian current element under the stated phasor convention.
\section{Three radial field scalings}
The electric field contains three characteristic powers of distance:
\begin{equation}
\boxed{
E\sim
\frac{1}{r^3}
+\frac{1}{r^2}
+\frac{1}{r}.
}
\end{equation}
They have different physical roles.
\subsection{The $1/r^3$ electrostatic-like term}
The most rapidly decaying term appears in the electric field and is tied to the oscillating charge separation. It resembles the static electric dipole field and dominates sufficiently close to the source, while still outside the idealized source itself.
\subsection{The $1/r^2$ induction term}
The $1/r^2$ terms occur in both electric and magnetic fields. They are associated with induction and with energy exchanged between the source and nearby fields. They decay faster than radiation and therefore become progressively less important with distance.
\subsection{The $1/r$ radiation term}
The $1/r$ terms are special because their power density falls as
\begin{equation}
\left(\frac{1}{r}\right)^2
=\frac{1}{r^2}.
\end{equation}
Multiplying by the area of a sphere,
\begin{equation}
4\pi r^2,
\end{equation}
produces a distance-independent total radiated power. This is exactly the geometric-spreading behavior developed earlier in EM17.
\begin{center}
\includegraphics{EM23_fig02_field_scaling_zones.png}
\vspace{0.4em}
\textit{Figure. Radial scaling of the three field families. The labels ``near'', ``intermediate'', and ``far'' describe dominant behavior, not infinitely sharp boundaries.}
\end{center}
\section{Near, intermediate, and far-field interpretation}
For the infinitesimal dipole, the dimensionless parameter controlling the hierarchy is
\begin{equation}
kr=\frac{2\pi r}{\lambda}.
\end{equation}
When
\begin{equation}
kr\ll1,
\end{equation}
the $1/r^3$ and $1/r^2$ terms are important. This is the reactive near-field regime of the idealized source.
When
\begin{equation}
kr\sim1,
\end{equation}
no single radial power need dominate. This is the transition or induction region.
When
\begin{equation}
kr\gg1,
\end{equation}
the $1/r$ terms dominate. This is the far-field or radiation regime for the point-source model.
For a real antenna of finite maximum dimension $D$, an additional far-field condition involving $D$ is required. A commonly used engineering criterion is
\begin{equation}
r\gtrsim\frac{2D^2}{\lambda},
\end{equation}
provided the observation point is also many wavelengths and many antenna dimensions away when those constraints are relevant \cite{Balanis2016,Stutzman2012}. The Hertzian element is idealized as $D\rightarrow0$, so the $kr$ hierarchy is the important one here.
\section{Far-field simplification}
In the far field, retain only the $1/r$ terms. Then
\begin{equation}
\boxed{E_r\approx0,}
\end{equation}
\begin{equation}
\boxed{
E_\theta^{\text{ff}}
=\frac{i\eta k I_0\ell}{4\pi r}
\sin\theta\,e^{-ikr},
}
\end{equation}
and
\begin{equation}
\boxed{
H_\phi^{\text{ff}}
=\frac{i k I_0\ell}{4\pi r}
\sin\theta\,e^{-ikr}.
}
\end{equation}
Therefore
\begin{equation}
\boxed{
\frac{E_\theta^{\text{ff}}}{H_\phi^{\text{ff}}}
=\eta.
}
\end{equation}
The far field has become locally plane-wave-like:
\begin{equation}
\mathbf E\perp\mathbf H,
\qquad
\mathbf E\perp\hat{\mathbf r},
\qquad
\mathbf H\perp\hat{\mathbf r}.
\end{equation}
Also,
\begin{equation}
\mathbf E\times\mathbf H
\parallel\hat{\mathbf r}.
\end{equation}
The source launches a spherical wave globally, but over a small patch of a sufficiently large sphere the wavefront is approximately planar.
\section{Why the near field is called reactive}
The radial powers alone do not tell the whole story. Phase matters too. Near-field electric and magnetic terms are not generally in the same phase. Energy is alternately stored in the electric and magnetic fields and can flow back toward the source during part of a cycle.
In contrast, the far-field terms share the same propagation phase and satisfy
\begin{equation}
E_\theta^{\text{ff}}=\eta H_\phi^{\text{ff}}.
\end{equation}
Their time-averaged Poynting vector is therefore outward and nonzero:
\begin{equation}
\langle\mathbf S\rangle
=\frac{1}{2}\Re\left\{
\mathbf E\times\mathbf H^*
\right\}.
\end{equation}
This is the physical distinction between merely having electromagnetic fields around a source and actually radiating net energy to infinity.
\section{Angular radiation pattern}
The far-field electric amplitude contains the angular factor
\begin{equation}
\boxed{\sin\theta.}
\end{equation}
Thus the field amplitude is zero on the dipole axis,
\begin{equation}
\theta=0,\pi,
\end{equation}
and maximum broadside to the dipole,
\begin{equation}
\theta=\frac{\pi}{2}.
\end{equation}
Because power is proportional to field amplitude squared, the normalized power pattern is
\begin{equation}
\boxed{
P_n(\theta)=\sin^2\theta.
}
\end{equation}
The pattern is independent of azimuth $\phi$. Rotating the two-dimensional figure-eight power cut about the $z$ axis produces the familiar doughnut-shaped three-dimensional radiation pattern.
\begin{center}
\includegraphics{EM23_fig03_dipole_radiation_pattern.png}
\vspace{0.4em}
\textit{Figure. Normalized Hertzian-dipole power pattern in a plane containing the dipole axis. The field is zero along the dipole and maximum broadside. Rotational symmetry about $z$ gives the three-dimensional doughnut pattern.}
\end{center}
\section{Far-field power density}
Using the far-field relation
\begin{equation}
H_\phi^{\text{ff}}
=\frac{E_\theta^{\text{ff}}}{\eta},
\end{equation}
the radial time-average Poynting vector is
\begin{equation}
\langle S_r\rangle
=\frac{|E_\theta^{\text{ff}}|^2}{2\eta}.
\end{equation}
Since
\begin{equation}
|E_\theta^{\text{ff}}|^2
=\frac{\eta^2k^2|I_0\ell|^2}{16\pi^2r^2}
\sin^2\theta,
\end{equation}
we obtain
\begin{equation}
\boxed{
\langle S_r\rangle
=\frac{\eta k^2|I_0\ell|^2}{32\pi^2r^2}
\sin^2\theta.
}
\end{equation}
The $1/r^2$ dependence of power density is now explicit.
\section{Radiation intensity}
A useful distance-independent quantity is the radiation intensity
\begin{equation}
\boxed{
U(\theta,\phi)
=r^2\langle S_r\rangle.
}
\end{equation}
For the Hertzian dipole,
\begin{equation}
\boxed{
U(\theta)
=\frac{\eta k^2|I_0\ell|^2}{32\pi^2}
\sin^2\theta.
}
\end{equation}
Its units are watts per steradian. Radiation intensity removes the trivial spherical-spreading factor and leaves only the angular distribution of radiated power.
\section{Total radiated power}
The total power passing through a large sphere is
\begin{equation}
P_{\text{rad}}
=\int_{4\pi}U(\theta,\phi)\,d\Omega.
\end{equation}
Using
\begin{equation}
d\Omega=\sin\theta\,d\theta\,d\phi,
\end{equation}
we have
\begin{align}
P_{\text{rad}}
&=\frac{\eta k^2|I_0\ell|^2}{32\pi^2}
\int_0^{2\pi}d\phi
\int_0^\pi\sin^3\theta\,d\theta.
\end{align}
The angular integrals are
\begin{equation}
\int_0^{2\pi}d\phi=2\pi
\end{equation}
and
\begin{equation}
\int_0^\pi\sin^3\theta\,d\theta=\frac{4}{3}.
\end{equation}
Therefore
\begin{equation}
\boxed{
P_{\text{rad}}
=\frac{\eta k^2|I_0\ell|^2}{12\pi}.
}
\end{equation}
In free space, use
\begin{equation}
\eta_0\approx120\pi\,\Omega
\end{equation}
and
\begin{equation}
k=\frac{2\pi}{\lambda}
\end{equation}
to obtain
\begin{equation}
\boxed{
P_{\text{rad}}
=40\pi^2|I_0|^2
\left(\frac{\ell}{\lambda}\right)^2.
}
\end{equation}
Here $I_0$ is the \emph{peak phasor current amplitude}; the factor $1/2$ associated with time averaging has already been included.
\begin{center}
\includegraphics{EM23_fig04_radiated_power_sphere.png}
\vspace{0.4em}
\textit{Figure. Total radiated power is obtained by integrating the outward far-field Poynting flux over any sufficiently large sphere. The local flux varies as $\sin^2\theta/r^2$, while the area element grows as $r^2\sin\theta\,d\theta\,d\phi$.}
\end{center}
\section{Radiation resistance}
It is convenient to represent the radiated power by an equivalent resistance defined through
\begin{equation}
P_{\text{rad}}
=\frac{1}{2}R_{\text{rad}}|I_0|^2.
\end{equation}
Comparing with the free-space radiated-power expression gives
\begin{equation}
\boxed{
R_{\text{rad}}
=80\pi^2
\left(\frac{\ell}{\lambda}\right)^2
\Omega.
}
\end{equation}
This is not an ohmic resistor. It is a circuit representation of real power carried away by electromagnetic radiation.
A terminology warning is important. The formula above is for the ideal \emph{Hertzian} or infinitesimal dipole with uniform current $I_0$ over length $\ell$. A physically short center-fed dipole has an approximately triangular current distribution and a different radiation-resistance coefficient. The two models should not be mixed.
\section{Directivity of the Hertzian dipole}
Although antenna gain will be developed systematically in EM24, the radiation pattern already lets us calculate directivity.
The maximum radiation intensity occurs at
\begin{equation}
\theta=\frac{\pi}{2},
\end{equation}
so
\begin{equation}
U_{\max}
=\frac{\eta k^2|I_0\ell|^2}{32\pi^2}.
\end{equation}
The average radiation intensity over all directions is
\begin{equation}
U_{\text{avg}}
=\frac{P_{\text{rad}}}{4\pi}.
\end{equation}
Therefore the maximum directivity is
\begin{equation}
D_{\max}
=\frac{U_{\max}}{U_{\text{avg}}}
=\frac{4\pi U_{\max}}{P_{\text{rad}}}.
\end{equation}
Substitution gives
\begin{equation}
\boxed{D_{\max}=\frac{3}{2}=1.5.}
\end{equation}
In decibels relative to isotropic,
\begin{equation}
\boxed{
D_{\max,\text{dBi}}
=10\log_{10}(1.5)
\approx1.76\,\text{dBi}.
}
\end{equation}
This is directivity, not yet gain. Gain will require radiation efficiency as well as angular concentration.
\section{Worked numerical example}
Consider an ideal Hertzian element in free space at
\begin{equation}
f=300\,\text{MHz}.
\end{equation}
The wavelength is
\begin{equation}
\lambda=\frac{c}{f}
\approx0.9993\,\text{m}.
\end{equation}
Let
\begin{equation}
\ell=\frac{\lambda}{50}
\approx1.999\times10^{-2}\,\text{m}
\end{equation}
and let the peak current amplitude be
\begin{equation}
|I_0|=1.00\,\text{A}.
\end{equation}
The radiation resistance is
\begin{align}
R_{\text{rad}}
&=80\pi^2\left(\frac{1}{50}\right)^2\\
&\approx0.3158\,\Omega.
\end{align}
Thus
\begin{equation}
\boxed{
P_{\text{rad}}
=\frac{1}{2}R_{\text{rad}}|I_0|^2
\approx0.1579\,\text{W}.
}
\end{equation}
Now observe the antenna broadside at
\begin{equation}
r=10\lambda,
\qquad
\theta=90^\circ.
\end{equation}
The far-field electric amplitude is
\begin{equation}
|E_\theta|
=\frac{\eta_0k|I_0|\ell}{4\pi r}
\approx0.3770\,\text{V/m}.
\end{equation}
The magnetic amplitude is
\begin{equation}
|H_\phi|
=\frac{|E_\theta|}{\eta_0}
\approx1.001\times10^{-3}\,\text{A/m}.
\end{equation}
The average local power density is
\begin{equation}
\langle S_r\rangle
=\frac{|E_\theta|^2}{2\eta_0}
\approx1.886\times10^{-4}\,\text{W/m}^2.
\end{equation}
The result is small locally because the radiated energy has spread over a large spherical area, but integration over all directions recovers the same $0.1579\,\text{W}$ total radiated power.
\section{General far-field current-distribution form}
The Hertzian element is a building block for arbitrary antennas. For a localized current distribution observed far away, write
\begin{equation}
R=|\mathbf r-\mathbf r'|.
\end{equation}
When
\begin{equation}
r\gg|\mathbf r'|,
\end{equation}
we use the far-zone approximations
\begin{equation}
\frac{1}{R}\approx\frac{1}{r}
\end{equation}
and
\begin{equation}
R\approx r-\hat{\mathbf r}\cdot\mathbf r'.
\end{equation}
Therefore
\begin{equation}
e^{-ikR}
approx
e^{-ikr}
e^{ik\hat{\mathbf r}\cdot\mathbf r'}.
\end{equation}
The vector potential becomes
\begin{equation}
\boxed{
\mathbf A_{\text{ff}}(\mathbf r)
\approx
\frac{\mu e^{-ikr}}{4\pi r}
\int
\mathbf J(\mathbf r')
e^{ik\hat{\mathbf r}\cdot\mathbf r'}
\,dV'.
}
\end{equation}
This expression contains one of the central ideas of antenna theory: radiation in a given direction is a coherent sum of contributions from the entire current distribution, each weighted by its relative propagation phase. Antenna patterns and phased arrays ultimately grow from this phase-summation principle.
\section{What changes between near field and far field}
Several properties emerge together in the far zone:
\begin{itemize}
\item the $1/r$ field terms dominate;
\item the radial electric component becomes negligible;
\item $\mathbf E$, $\mathbf H$, and $\hat{\mathbf r}$ become mutually perpendicular;
\item the field ratio approaches the medium impedance $\eta$;
\item electric and magnetic far fields are in phase for a lossless medium;
\item the average Poynting vector is directed radially outward;
\item angular pattern becomes independent of distance;
\item power density falls as $1/r^2$.
\end{itemize}
These are precisely the properties needed to connect Maxwell's equations to practical antenna quantities such as radiation intensity, directivity, gain, effective aperture, EIRP, and eventually the Friis transmission equation.
\section{Common mistakes}
\begin{itemize}
\item \textbf{Calling every $1/r^2$ quantity a near field.} Far-field \emph{power density} falls as $1/r^2$ even though the far-field \emph{field amplitudes} fall as $1/r$.
\item \textbf{Assuming the near field carries no instantaneous power.} Energy can flow locally in either direction. The distinction is that reactive energy is returned rather than producing a distance-independent net radiated power.
\item \textbf{Dropping $E_r$ everywhere.} The radial electric field is negligible in the far zone, not in the general Hertzian-dipole solution.
\item \textbf{Using a far-field formula at $kr\ll1$.} The discarded $1/r^2$ and $1/r^3$ terms can then dominate.
\item \textbf{Confusing field pattern with power pattern.} The Hertzian field amplitude is proportional to $\sin\theta$; the power pattern is proportional to $\sin^2\theta$.
\item \textbf{Mixing peak and RMS current conventions.} Here $I_0$ is a peak phasor amplitude, so $P=(1/2)R|I_0|^2$.
\item \textbf{Confusing Hertzian and physically short dipoles.} Their current distributions, and therefore their radiation-resistance coefficients, differ.
\end{itemize}
\section{What EM23 adds to the series}
EM16--EM21 established electromagnetic waves, energy flow, propagation, polarization, and boundary behavior. EM22 showed how those waves can be guided on transmission lines. EM23 now makes the transition from guided waves to \emph{radiation from sources}.
The key field results for the ideal $z$-directed Hertzian element are
\begin{equation}
\boxed{
H_\phi
=\frac{I_0\ell\sin\theta}{4\pi}
\left(
\frac{ik}{r}+\frac{1}{r^2}
\right)e^{-ikr},
}
\end{equation}
\begin{equation}
\boxed{
E_r
=\frac{\eta I_0\ell\cos\theta}{2\pi}
\left(
\frac{1}{r^2}-\frac{i}{kr^3}
\right)e^{-ikr},
}
\end{equation}
and
\begin{equation}
\boxed{
E_\theta
=\frac{\eta I_0\ell\sin\theta}{4\pi}
\left(
\frac{ik}{r}+\frac{1}{r^2}-\frac{i}{kr^3}
\right)e^{-ikr}.
}
\end{equation}
In the radiation zone,
\begin{equation}
\boxed{
E_\theta^{\text{ff}}
=\eta H_\phi^{\text{ff}}
\propto
\frac{\sin\theta}{r},
}
\end{equation}
which leads to
\begin{equation}
\boxed{
U(\theta)\propto\sin^2\theta,
}
\end{equation}
\begin{equation}
\boxed{
P_{\text{rad}}
=\frac{\eta k^2|I_0\ell|^2}{12\pi},
}
\end{equation}
and in free space
\begin{equation}
\boxed{
R_{\text{rad}}
=80\pi^2
\left(\frac{\ell}{\lambda}\right)^2\Omega.
}
\end{equation}
The next article can build directly on these results to define radiation intensity, beam solid angle, directivity, radiation efficiency, antenna gain, EIRP, and effective aperture systematically.
\section*{References}
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C. A. Balanis,
\emph{Antenna Theory: Analysis and Design}, 4th ed.,
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\emph{Antenna Theory and Design}, 3rd ed.,
Wiley, 2012.
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D. J. Griffiths,
\emph{Introduction to Electrodynamics}, 4th ed.,
Cambridge University Press, 2017.
\bibitem{Jackson1999}
J. D. Jackson,
\emph{Classical Electrodynamics}, 3rd ed.,
Wiley, 1999.
\bibitem{Ramo1994}
S. Ramo, J. R. Whinnery, and T. Van Duzer,
\emph{Fields and Waves in Communication Electronics}, 3rd ed.,
Wiley, 1994.
\bibitem{Ulaby2015}
F. T. Ulaby and U. Ravaioli,
\emph{Fundamentals of Applied Electromagnetics}, 7th ed.,
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\end{thebibliography}