---
title: Dipole Radiation
module: Maxwell’s Equations and Electromagnetic Waves
moduleNumber: 10
lessonNumber: 4
order: 1004
summary: >
  Only accelerating charge radiates, and the simplest accelerator is a charge sloshing back
  and forth: an oscillating electric dipole. We work out the field it throws off, keeping
  the part that survives to large distance — the $1/r$ radiation field whose intensity goes
  as $\sin^2\theta/r^2$, zero along the dipole axis and strongest broadside. From it follow
  the $\omega^4$ scaling of total radiated power, radiation resistance as the feed's view of
  that escaping power, and, through reciprocity, the fact that a good transmitter receives
  well in the same directions. The near-zone terms that fall off faster carry no net power,
  and we mark carefully where each description is allowed to be used.
topics: [Maxwell’s Equations and Electromagnetic Waves]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 30 — Maxwell’s Equations and Electromagnetic Waves; §30-4 Electromagnetic Radiation"
---

## Accelerated charge and outward disturbance

An electromagnetic wave originates when charges accelerate. A stationary charge has a
static electric field. A charge moving at constant velocity has a transformed but
nonradiative field pattern in every inertial description. Acceleration changes the
source field pattern. Information about that change travels outward at the speed of
light, carrying a transverse field component that can remain finite when multiplied by
distance from the source.

An antenna is a controlled collection of accelerating charges. In a straight driven
conductor, current reverses every half cycle. Charge accumulates at one end, returns
through the feed, and accumulates at the other end. The dipole moment reverses with the
drive. The surrounding electric and magnetic fields cannot adjust everywhere at one
instant. The changed field configuration occupies an outward-moving region. At a large
distance it becomes a freely propagating electromagnetic wave.

The causal delay is written with the retarded time

$$
t_r=t-\frac{r}{c}.
$$

Here $r$ is the distance from the source region to an observation point. A source
quantity evaluated at $t_r$ gives its value when the emitted disturbance began
propagating toward the observation point. Retarded time fixes the wave phase, the shell
structure of a pulse, and the relation between source motion and receiver signal.

A compact source has distance dependence that separates the field into three scales.
Terms proportional to $1/r^3$ dominate very near the source and resemble an
instantaneous electric-dipole field with a retarded source value. Terms proportional to
$1/r^2$ describe induction effects. Terms proportional to $1/r$ are the radiation
fields. Squaring a $1/r$ field gives a power flux proportional to $1/r^2$, which remains
compatible with the same total power crossing every spherical surface around the source.

The labels "near zone" and "far zone" require a wavelength as well as a distance. For a
short antenna of size $D\ll\lambda$, the radiation term dominates when $r$ is much
larger than a wavelength and much larger than $D$. The reactive near region lies at
distances small compared with a wavelength. Between them, induction and radiation terms
can be comparable. The boundaries are gradual; they are not physical surfaces in space.

The near field stores and returns energy to the source during a cycle. It does not have
the simple local relation $E=cB$ associated with a plane wave. Its electric and magnetic
parts can be out of phase, and its spatial directions have radial as well as transverse
components. A receiver placed in that region may couple capacitively or inductively to
the source rather than sample its radiated far-field pattern. Measurements made there
cannot be interpreted by applying a far-field $1/r^2$ intensity law.

In the radiation zone, the fields are transverse to the radial propagation direction.
Their leading components are in phase, satisfy $E=cB$ in magnitude, and have a radial
time-averaged Poynting vector. The energy-flow and radiation-pressure relations belong
to [Electromagnetic Momentum](/electricity-and-magnetism/maxwell-electromagnetic-waves/electromagnetic-momentum). The present lesson uses only the local radiation-zone intensity needed for antenna patterns and receiver estimates.

$$
% caption: Near-zone versus far-zone character. Close to the source the field has radial as well as transverse parts and stores energy reactively; far from the source the leading E and B are transverse, mutually perpendicular, satisfy E=cB in magnitude, and carry energy outward.
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$$

### The oscillating electric dipole

The elementary antenna model has a time-dependent electric dipole moment along a fixed
axis. Let the axis be $z$ and write

$$
\vec p(t)=p(t)\,\hat z.
$$

A pair of charges $+q$ and $-q$ separated by vector $\vec\ell$ has moment
$\vec p=q\vec\ell$. In a metal antenna, charge is distributed along the
conductors rather than concentrated at two mathematical points. The dipole moment still
summarizes the lowest-order radiating charge separation when the antenna is short
compared with the wavelength.

$$
% caption: One half-cycle of an oscillating electric dipole. Charge piles up at opposite ends a quarter-period apart; the rod current is largest at the middle phase, when the end charges pass through zero.
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For harmonic drive, use

$$
p(t)=p_0\cos\omega t,
\qquad
\omega=2\pi f,
\qquad
\lambda=\frac{c}{f}.
$$

The emitted radiation has the drive frequency $f$ and the corresponding free-space
wavelength $\lambda$. A pulse or a nonsinusoidal drive has a range of frequency
components. Antenna dimensions, matching networks, and the receiving circuit select
different portions of that spectrum. In a uniform material, replace $c$ by the phase
velocity for the wavelength calculation while the frequency remains fixed across the
boundary.

The radiation strength depends on acceleration. For a sinusoidal dipole, the second
time derivative has amplitude $\omega^2p_0$. Doubling frequency at fixed dipole-moment
amplitude doubles the charge acceleration twice over and produces a far-field amplitude
four times larger. The radiated power rises as the fourth power of frequency at fixed
$p_0$. This scaling holds for the short-dipole model, before feed current, antenna size,
and resonance vary with frequency.

## Radiation-zone fields of a short dipole

Take an observation point at radius $r$ and polar angle $\theta$ from the dipole axis.
The leading radiation-zone electric field has the following magnitude when source size
is small compared with both $r$ and $\lambda$:

$$
|E_\theta(r,\theta,t)|
=\frac{1}{4\pi\varepsilon_0c^2}
\frac{|\ddot p(t-r/c)|}{r}\sin\theta.
$$

The corresponding magnetic field is azimuthal and has magnitude

$$
|B_\phi|=\frac{|E_\theta|}{c}.
$$

The expression identifies four independent controls. Field magnitude decreases as
$1/r$. It vanishes on the dipole axis because $\sin\theta=0$. It reaches a maximum in
the plane normal to that axis. Its time dependence follows the source acceleration at
the retarded time. The formula describes a radiation-zone component; it does not give
the total electric field close to the antenna.

$$
% caption: Radiation-zone geometry for a short dipole on the vertical axis. The observation direction is radial at polar angle theta from the axis; the radiated electric field is transverse in the plane containing the axis and the magnetic field circles the axis.
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$$

The radiation-zone field has no radial electric component at leading order. The radial
and induction components fall faster with distance and become negligible compared with
the transverse field at a sufficiently large radius. That distinction is visible in a
measurement. A small electric probe oriented radially near an antenna can register a
large signal, while the same orientation far away approaches a null. A far-zone probe
should be oriented for the expected transverse electric field.

A harmonic moment has RMS radiation-zone field

$$
E_{\theta,\mathrm{rms}}
=\frac{p_0\omega^2}{4\pi\varepsilon_0c^2r\sqrt2}\sin\theta.
$$

The RMS radial intensity follows from the local plane-wave relation:

$$
\left\langle S_r\right\rangle
=\frac{E_{\theta,\mathrm{rms}}^2}{\mu_0c}
=\frac{p_0^2\omega^4}{32\pi^2\varepsilon_0c^3r^2}\sin^2\theta.
$$

The angular dependence is the characteristic electric-dipole pattern. The power per
unit solid angle is proportional to $\sin^2\theta$. The pattern is axially symmetric:
rotating an observation point around the dipole axis does not change the intensity at a
fixed polar angle. The axis contains two nulls. The plane perpendicular to the axis is
a continuous ring of maximum intensity.

$$
% caption: Polar section of the short-dipole power pattern. The radius in each direction is proportional to sin^2(theta), giving two broadside lobes and a null along the antenna axis.
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$$

The same pattern can be viewed in three dimensions. Revolving the two-dimensional
cross-section about the dipole axis produces a torus-like radiation lobe. The word
"torus" describes the constant-intensity surface, not a separate waveguide or ring
source. Every direction in the broadside plane has the same ideal intensity. Real
antennas depart from this symmetry because of finite length, nearby conductors, ground,
feed geometry, and material loss.

$$
% caption: Three-dimensional radiation pattern of a short dipole. Revolving the broadside lobe about the antenna axis sweeps out a torus, with deep nulls along the axis above and below the antenna.
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$$

The intensity formula can be converted into a differential power distribution by
multiplying by $r^2$:

$$
\frac{dP}{d\Omega}
=r^2\left\langle S_r\right\rangle
=\frac{p_0^2\omega^4}{32\pi^2\varepsilon_0c^3}\sin^2\theta.
$$

The factor $d\Omega$ is a small solid angle subtended at the antenna. Integrating over
all directions uses

$$
\int_0^{2\pi}\int_0^\pi\sin^2\theta\sin\theta\,\d\theta\,\d\phi
=\frac{8\pi}{3}.
$$

The total time-averaged radiated power of the ideal oscillating electric dipole is

$$
P_{\mathrm{rad}}
=\frac{p_0^2\omega^4}{12\pi\varepsilon_0c^3}.
$$

No radius appears in the final result because the weaker field at a larger radius is
spread across a proportionally larger sphere. The total power is finite only after the
full angular distribution is included. Replacing the dipole pattern by a uniform sphere
would preserve neither the axial null nor the broadside maximum.

### Frequency and size scaling

The $\omega^4$ factor in $P_{\mathrm{rad}}$ belongs to a fixed dipole-moment amplitude.
An antenna is often specified by feed current and physical length instead. For an ideal
short current element with uniform current amplitude $I_0$, charge amplitude is
approximately $I_0/\omega$, so the dipole-moment amplitude is

$$
p_0\approx\frac{I_0\ell}{\omega},
\qquad \ell\ll\lambda.
$$

Substitution gives a power proportional to $I_0^2\ell^2\omega^2$. With RMS current,
the standard short-dipole radiation resistance is

$$
R_{\mathrm{rad}}=80\pi^2\left(\frac{\ell}{\lambda}\right)^2,
\qquad
P_{\mathrm{rad}}=I_{\mathrm{rms}}^2R_{\mathrm{rad}}.
$$

Radiation resistance is an equivalent resistance assigned at the feed. It accounts for
power leaving as electromagnetic radiation, not heat in a conductor. The feed sees it
as a real-power path. Ohmic resistance, dielectric loss, ground loss, and matching
network loss add separate dissipative terms. The radiation efficiency is the fraction
of accepted power associated with $R_{\mathrm{rad}}$ rather than those loss resistances.

$$
% caption: Short-dipole radiation resistance grows as the square of electrical length. A rod of fixed physical length radiates more strongly as its length becomes a larger fraction of the wavelength.
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$$

The formula is restricted to $\ell\ll\lambda$. A half-wave dipole has a nonuniform
standing current and a different radiation resistance and pattern. Its broadside
maximum and axial null remain recognizable, but its angular distribution is narrower
than the short-dipole result. The electrical size $\ell/\lambda$, rather than metre
length alone, determines which model is appropriate. A 0.30 m conductor is electrically
short at 10 MHz and much larger at 1 GHz.

At fixed current and short-dipole length, radiated power rises as $f^2$ because
$R_{\mathrm{rad}}\propto(\ell/\lambda)^2\propto f^2$. At fixed dipole moment,
radiated power rises as $f^4$. These are different constraints. A calculation must
state whether charge separation amplitude, feed current, feed voltage, or accepted
power is being held fixed while frequency changes. Treating all of them as constant
would violate the antenna's circuit relations.

The far-zone radiation field at a fixed polar angle is proportional to $p_0\omega^2/r$.
This form supports a distance check. Doubling $r$ halves electric-field amplitude and
reduces intensity to one quarter. A response that falls much faster or changes direction
with probe orientation may be near-field coupling, a reflecting environment, a cable
current, or an antenna pattern null rather than free-space radial spreading.

## Electric-dipole reception

An electric-dipole receiving antenna responds to the electric field component along its
effective-height vector. The incident field drives charge toward one end of the
conductor and away from the other. With the feed open, this charge separation produces
an open-circuit voltage. With a receiver attached, the antenna current and delivered
voltage depend on the antenna impedance, the receiver impedance, and losses in the
conductors and matching network.

A small receiving dipole in a locally uniform far-zone electric field has

$$
V_{\mathrm{oc,rms}}
=\vec h_{\mathrm{eff}}\cdot\vec E_{\mathrm{rms}}
=h_{\mathrm{eff}}E_{\mathrm{rms}}\cos\alpha.
$$

The angle $\alpha$ is between the antenna effective-height vector and the incoming
electric-field vector. The scalar $h_{\mathrm{eff}}$ is the effective height. It is an
antenna property defined by received voltage per incident electric-field amplitude; it
is not generally the same as the physical rod length. Current distribution, end loading,
frequency, ground, and nearby conductors affect it.

If the rod is perpendicular to the electric field, an ideal short dipole has zero
open-circuit response in this approximation. A real receiver can still display a
residual signal from cross-polar response, cable pickup, multipath, finite support
structure, or imperfect alignment. A small nonzero reading should be separated from
the main co-polar response by calibration rather than assigned automatically to the
ideal antenna model.

The voltage response varies as $\cos\alpha$. If the receiver is connected to a
resistive matched input and all other conditions are fixed, the received power varies as
$\cos^2\alpha$. This orientation factor is one part of polarization matching. It does
not replace the transmitting antenna's radiation pattern: a receiver can be correctly
aligned with the local electric field and still be placed near a transmit null.

$$
% caption: A receiving dipole responds to the component of the incident electric field along the rod. The open-circuit voltage follows the cosine of the tilt angle alpha between the rod and E, and the received power its square.
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$$

### Antenna terminal model

At one frequency, a receiving antenna can be represented at its feed terminals by a
Thevenin source $V_{\mathrm{oc}}$ in series with an antenna impedance

$$
Z_A=R_{\mathrm{rad}}+R_{\mathrm{loss}}+jX_A.
$$

$R_{\mathrm{rad}}$ represents reradiation from current in the receive antenna.
$R_{\mathrm{loss}}$ represents conductor, dielectric, and ground losses. $X_A$ is the
net reactive part at the reference plane. A receiver or matching network attached to
the terminals has impedance $Z_L$. The terminal voltage is

$$
\widetilde V_L
=\widetilde V_{\mathrm{oc}}\frac{Z_L}{Z_A+Z_L}.
$$

The open-circuit voltage is a field-to-terminal conversion. The loaded voltage is a
circuit result. Confusing them overstates the receiver signal whenever the load draws
appreciable current. A high-input-impedance field probe seeks to preserve the open-
circuit condition. A matched receiver seeks available power and deliberately loads the
antenna.

$$
% caption: Thevenin equivalent of a receiving antenna. The incident wave appears as an open-circuit source in series with the antenna impedance; the delivered terminal signal depends on how the receiver load compares with that impedance.
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A lossless antenna with purely resistive input resistance $R_A$ delivers maximum
available real power to a matched resistive receiver $R_L=R_A$. If all voltages
are RMS values,

$$
P_{\mathrm{avail}}=\frac{V_{\mathrm{oc,rms}}^2}{4R_A}.
$$

With a complex antenna impedance, conjugate matching cancels the antenna reactance at
the chosen frequency and matches the resistive part. Matching does not create signal
power. It selects a load condition that transfers the available antenna power rather
than reflecting it back toward the feed. Receiver noise figure, bandwidth, and overload
limit remain independent constraints.

The received voltage is also controlled by the transmitter geometry. In the far zone of
a vertical short transmitting dipole, a receiver placed in the horizontal broadside
plane sees a vertical electric field. A vertical receiving dipole aligns with it. A
receiver on the transmitter axis lies in the ideal radiation null, so changing only the
receiver orientation cannot restore a far-zone signal. Move away from the null or use a
different transmitting geometry.

### Loop-antenna reception

A loop antenna responds primarily to the magnetic field through Faraday's law. For a
loop of $N$ turns, area $A$, and unit normal $\hat n$, a spatially uniform
magnetic field produces flux $NBA\cos\beta$, where $\beta$ is the angle between the
field and the loop normal. An electrically small loop in a sinusoidal plane wave has
RMS induced emf

$$
\mathcal E_{\mathrm{rms}}
=N A\omega B_{\mathrm{rms}}|\cos\beta|
=N A\frac{2\pi f}{c}E_{\mathrm{rms}}|\cos\beta|.
$$

The approximation requires the incident magnetic field to be nearly uniform across the
loop. It also requires the loop to be small enough that propagation phase across its
area and lead geometry can be neglected. A multi-turn loop increases induced emf in
proportion to $N$, but also increases resistance, inductance, stray capacitance, and
self-resonance effects.

$$
% caption: A small loop antenna responds to the magnetic field through Faraday's law. Coupling is greatest when the loop normal is parallel to B; the changing flux drives an emf and current around the loop. Turning the loop a right angle nulls it.
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$$

In a wave traveling in one direction, the magnetic field lies perpendicular to both
the propagation direction and the electric field. A loop receives most strongly when
its plane is perpendicular to $B$, equivalently when its normal points along $B$. Turn
the loop by a right angle and the magnetic flux approaches zero. This sharp null
supports direction finding when the incoming
magnetic field lies in its plane.

An electric dipole and a loop can distinguish different field components, but a
practical receiver includes both antenna and circuit response. The electric dipole
converts local electric field into an open-circuit terminal voltage. The loop converts
time-varying magnetic flux into emf. At the same frequency and range, the two output
voltages can differ by orders of magnitude because their effective height, area, turn
count, orientation, impedance, and receiver loading differ.

The loop formula becomes inaccurate when the loop diameter is a substantial fraction of
the wavelength. Different portions of the loop then experience different wave phase and
field direction. The flux integral must be evaluated with the spatially varying field,
and the loop behaves as a distributed antenna rather than a lumped Faraday loop. A
small-loop calculation should therefore state its diameter-to-wavelength ratio.

## Far-field pattern mapping and signal limits

An antenna pattern is a measured or calculated angular function at a stated frequency,
polarization, and range. For the ideal short dipole, the normalized power pattern is

$$
F_P(\theta)=\frac{\langle S_r(\theta)\rangle}{\langle S_r(\pi/2)\rangle}
=\sin^2\theta.
$$

The corresponding normalized electric-field amplitude pattern is $F_E(\theta)=|\sin
\theta|$. A power detector, a spectrum analyzer referred to a fixed impedance, and a
calibrated field meter usually report quantities proportional to power. A voltage probe
reports an amplitude. The distinction changes a pattern plotted in decibels: power
ratios use $10\log_{10}$, while voltage ratios use $20\log_{10}$ only when the same
reference impedance applies.

$$
% caption: Angular dependence of the field and power patterns. The far-zone electric-field amplitude follows |sin(theta)| while the intensity and received power follow sin^2(theta), so the power pattern is the narrower of the two.
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The measurement range must be in the transmitting antenna's far field. An antenna or
aperture with largest dimension $D$ has the common far-field planning rule

$$
r_F\gtrsim\frac{2D^2}{\lambda}.
$$

This criterion limits curvature of the outgoing wavefront across the receiving aperture.
The measurement range must also span many wavelengths from the source and satisfy the
short-dipole radiation-zone condition when that model is used. A small dipole can reach
its radiation-zone behavior at a shorter range than a large aperture, while a large
array may require a much longer range than the simple $r\gg\lambda$ statement.

$$
% caption: Far-field range planning. The observation distance must be large enough, roughly r > 2D^2/lambda, that the spherical wavefront is nearly planar across the receiving aperture of size D.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
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\draw[black,very thick] (1.00,.95)--(1.00,2.15);
\node[below] at (1.00,.80) {send};
\draw[acc,thick,dashed] (3.55,.85) .. controls (3.88,1.55) .. (3.55,2.35);
\draw[acc,thick] (4.55,.75) .. controls (4.92,1.55) .. (4.55,2.55);
\draw[black,very thick] (5.10,.80)--(5.10,2.50);
\node[below] at (5.10,.60) {aperture};
\draw[<->,black,thick] (1.20,1.55)--(4.85,1.55) node[midway,below] {range};
\draw[<->,black,thick] (5.42,.80)--(5.42,2.50) node[midway,right] {$D$};
\end{tikzpicture}
$$

Pattern mapping fixes the transmitter and keeps its drive frequency, output level, and
polarization stable. The receiver is placed at a fixed radius and rotated through a
defined angular coordinate. A vertical dipole can be mapped in an elevation plane by
changing the polar angle from the vertical axis. An azimuth scan rotates around that
axis. Record which coordinate changed, the receiver polarization, the reference
direction, and the measurement plane. A polar plot without those labels is not enough
to reproduce an antenna pattern.

Normalize each stable data set to its maximum value:

$$
P_N(\theta)=\frac{P_R(\theta)}{P_{R,\max}}.
$$

Use a linear plot near nulls when their absolute depth matters. A decibel plot displays
side response and weak cross-polar pickup that a linear plot can obscure. A detector with an
unknown logarithmic response should be calibrated before data are normalized; otherwise
the apparent pattern shape can be an instrument transfer curve rather than an antenna
property.

### Environmental and receiver limits

Ground, walls, support poles, people, test equipment, and the receiver cable can alter
a measured pattern. A reflected path adds to the direct path with a phase set by path
length and reflection coefficient. As the receiver moves through angle, the two paths
can add or cancel. The result can look like an antenna lobe, especially when the range
is only a few wavelengths above a ground plane. An anechoic range reduces reflections;
an outdoor range requires documented ground geometry and enough height or distance to
separate direct and reflected paths.

The feed cable should be routed so that it has little unintended coupling to the
antenna. A common-mode current on an outer conductor can radiate or receive, changing
the pattern of the assembly. Ferrite suppression, balanced feeds, chokes, cable routing,
and repeat scans with a changed cable path help distinguish the intended antenna response
from cable response. The test report should identify the reference plane at which power
or voltage was measured.

Every receiver has a noise floor and a finite linear range. Below the noise floor,
displayed values are dominated by receiver noise, external background, or averaging
statistics. Above the linear range, a front end or detector can compress and reduce the
apparent lobe contrast. Use a reference source level that keeps the broadside maximum
below compression while leaving the expected null depth above the measurement floor.
Repeat a subset of angles at a second source level. A normalized curve that changes
shape with source level indicates a receiver nonlinearity or an interference problem.

Distance error, angular error, polarization mismatch, calibration error, and source
drift should be tracked separately. In the far field, a fixed-direction received power
scales approximately as $r^{-2}$, so a small range uncertainty gives

$$
\frac{\delta P_R}{P_R}\approx2\frac{\delta r}{r}.
$$

For independent fractional uncertainties, a planning estimate uses root-sum-square
combination:

$$
u_{\mathrm{rel}}
=\sqrt{u_{\mathrm{cal}}^2+u_{\mathrm{range}}^2+u_{\mathrm{angle}}^2+u_{\mathrm{drift}}^2}.
$$

This expression summarizes independent random terms; systematic-error searches remain
separate. A ground reflection can shift every angular point in a correlated way and
will not be revealed by averaging repeated readings at the same geometry. Change
range, height, polarization, or cable path to expose such effects.

## Worked calculations and design checks

> **Worked example (short-dipole radiation resistance and power).** An ideal short current
> element of length $\ell=0.150\ \mathrm{m}$ is driven at $f=100\ \mathrm{MHz}$. The
> free-space wavelength is
>
> $$
> \lambda=\frac{3.00\times10^8\ \mathrm{m\,s^{-1}}}{1.00\times10^8\ \mathrm{s^{-1}}}
> =3.00\ \mathrm{m},
> $$
>
> so the electrical length is $\ell/\lambda=0.050$ — small enough for a first short-dipole
> estimate. The radiation resistance is
>
> $$
> R_{\mathrm{rad}}
> =80\pi^2(0.050)^2
> =1.97\ \Omega,
> $$
>
> and at $I_{\mathrm{rms}}=0.200\ \mathrm{A}$ the ideal radiated power is
>
> $$
> P_{\mathrm{rad}}
> =(0.200\ \mathrm{A})^2(1.97\ \Omega)
> =7.90\times10^{-2}\ \mathrm{W}.
> $$
>
> Adding a loss resistance $R_{\mathrm{loss}}=0.80\ \Omega$, the accepted real power is
> $I_{\mathrm{rms}}^2(R_{\mathrm{rad}}+R_{\mathrm{loss}})=0.111\ \mathrm{W}$ and the radiation
> efficiency is
>
> $$
> \eta_{\mathrm{rad}}
> =\frac{R_{\mathrm{rad}}}{R_{\mathrm{rad}}+R_{\mathrm{loss}}}
> =\frac{1.97}{1.97+0.80}
> =0.711.
> $$
>
> About 71 percent of the accepted real power is radiated; the rest heats the antenna.

Feedline loss before the antenna and mismatch loss between a
source and the feed are additional power-budget entries. A transmitter output rating
cannot be substituted directly for accepted antenna power without those entries.

> **Worked example (combined angle and range factors).** At a fixed range, the dipole
> intensity relative to broadside is $\sin^2\theta$. At $\theta=30^\circ$,
>
> $$
> \frac{I(30^\circ)}{I(90^\circ)}=\sin^2(30^\circ)=0.250,
> $$
>
> one quarter of the broadside value, while the electric-field amplitude ratio is
> $\sin(30^\circ)=0.500$: a voltage detector reads half, a power detector a quarter. A
> receiving rod also misaligned by $30^\circ$ with the local electric field contributes a
> further power factor $\cos^2(30^\circ)=0.750$. Doubling the range from $r$ to $2r$ adds
> another factor $1/4$ in intensity, so
>
> $$
> \frac{P_R(30^\circ,2r)}{P_R(90^\circ,r)}
> =\left(\frac14\right)\left(\frac14\right)(0.750)
> =0.0469,
> $$
>
> about 4.69 percent of the broadside, aligned, shorter-range power. This assumes free-space
> far-field propagation and a matched receiver, with no ground reflection or cable pickup.

> **Worked example (small-loop induced emf).** A single-turn loop of radius
> $a=0.100\ \mathrm{m}$ receives a plane wave with $E_{\mathrm{rms}}=0.150\ \mathrm{V\,m^{-1}}$,
> its plane perpendicular to the magnetic field so $|\cos\beta|=1$. The loop area is
> $A=\pi a^2=3.14\times10^{-2}\ \mathrm{m^2}$. At $f=600\ \mathrm{kHz}$, using
> $B_{\mathrm{rms}}=E_{\mathrm{rms}}/c$ and $\omega=2\pi f$,
>
> $$
> \mathcal E_{\mathrm{rms}}
> =A\omega\frac{E_{\mathrm{rms}}}{c}
> =(3.14\times10^{-2})
> \frac{2\pi(6.00\times10^5)}{3.00\times10^8}(0.150)
> =5.92\times10^{-5}\ \mathrm{V},
> $$
>
> that is $59.2\ \mathrm{\mu V}$, not millivolts. At $60.0\ \mathrm{MHz}$ — one hundred times
> the frequency with the other quantities unchanged — the emf rises to $5.92\ \mathrm{mV}$.
> The corresponding wavelengths, 500 m and 5.00 m, both far exceed the 0.200 m loop diameter,
> so the uniform-field approximation holds; a loaded receiver reads less than this open-loop
> emf according to its terminal impedance.

> **Worked example (far-field range and uncertainty budget).** An antenna under test has
> largest dimension $D=0.80\ \mathrm{m}$ at 300 MHz, so $\lambda=1.00\ \mathrm{m}$ and the
> aperture-based range estimate is
>
> $$
> r_F\approx\frac{2(0.80\ \mathrm{m})^2}{1.00\ \mathrm{m}}
> =1.28\ \mathrm{m}.
> $$
>
> This is a lower bound, not a sufficient outdoor range: a 6 m range is six wavelengths and
> leaves room to separate the direct and reflected paths. For a normalized power sample with
> calibration uncertainty 5.0 percent, range uncertainty $\delta r=0.050\ \mathrm{m}$ at
> $r=10.0\ \mathrm{m}$, and source drift 2.0 percent, the range term is
>
> $$
> u_{\mathrm{range}}=2\frac{0.050}{10.0}=0.010=1.0\%,
> $$
>
> and the independent relative uncertainty is
>
> $$
> u_{\mathrm{rel}}
> =\sqrt{(0.050)^2+(0.010)^2+(0.020)^2}
> =0.0548.
> $$
>
> Report the point as a normalized power with roughly a 5.5 percent random-error budget, and
> list reflection or polarization concerns separately.

A compact record
contains frequency, transmitter level, range, receiver orientation, cable routing,
detector bandwidth, reference direction, data normalization, and the environmental
conditions at the time of the scan.

### Calculation audit

Several quick checks catch most antenna-radiation mistakes:

- **Zone check.** Verify that the observation range supports the radiation-zone model
  before using $E=cB$, $1/r$ field scaling, or an angular far-field pattern.
- **Angle check.** Measure polar angle from the dipole axis. Broadside is
  $\theta=90^\circ$, not $\theta=0$.
- **Quantity check.** Keep field amplitude, intensity, received voltage, and received
  power separate. Their distance and alignment exponents differ.
- **RMS check.** Use RMS quantities consistently in receiver-power and loop-emf
  calculations. A peak voltage inserted into an RMS power formula creates a factor-of-
  two error.
- **Model check.** State whether the source is a short dipole, a finite resonant dipole,
  or a measured antenna. The $\sin^2\theta$ and radiation-resistance formulas have
  stated limits.

These checks keep the radiation model tied to an actual geometry, frequency, and
measurement reference. They also separate antenna behavior from receiver behavior,
which is necessary when a weak signal or an unexpected lobe appears in an experiment.

## Effective aperture, reciprocity, and frequency response

The electric-dipole receive model can also be expressed as an effective aperture. For a
plane wave of time-averaged intensity $S$ arriving from the antenna's maximum-response
direction with matching polarization and conjugate terminal match,

$$
P_{\mathrm{avail}}=S A_e.
$$

$A_e$ is an effective area. It describes the available received power, not the literal
metal cross-sectional area of a wire. A resonant or electrically short antenna can have
an effective aperture larger than the visible wire area because the incident wave drives
coherent current throughout the conducting structure. The energy is supplied by the
incident field over the effective capture region; the antenna does not create it.

A reciprocal antenna in free space obeys

$$
A_e=\frac{G\lambda^2}{4\pi},
$$

where $G$ is gain in the selected direction. Gain includes radiation efficiency. A
lossless ideal short dipole has directivity $D=1.5$ and gain $G=1.5$. Loss resistance
reduces gain below directivity by the radiation-efficiency factor. The formula applies to
a stated polarization and impedance match; a polarization mismatch or a receiver
mismatch reduces delivered power below $SA_e$.

> **Worked example (effective aperture of a short dipole).** A lossless short dipole at
> 100 MHz has $\lambda=3.00\ \mathrm{m}$ and gain $G=1.5$, so its effective aperture is
>
> $$
> A_e=\frac{(1.5)(3.00\ \mathrm{m})^2}{4\pi}=1.07\ \mathrm{m^2}.
> $$
>
> An incident intensity of $1.00\ \mathrm{\mu W\,m^{-2}}$ then delivers a matched available
> power of $1.07\ \mathrm{\mu W}$ before receiver loss, provided the wave arrives broadside
> with its electric field parallel to the dipole. Off broadside the aperture carries the
> pattern factor $\sin^2\theta$, and a linear-polarization misalignment $\alpha$ multiplies
> it by $\cos^2\alpha$.

Reciprocity connects the transmitting and receiving descriptions. A passive linear
antenna has the same directional pattern and polarization response when it transmits as
when it receives at the same frequency. A short dipole that has broadside transmission
maximum also has broadside receiving maximum. The transmit null on the rod axis is the
receive null for an incident field with the corresponding polarization. Reciprocity
does not state that transmitter and receiver powers are equal; it connects normalized
directional response after their source and load conditions are specified.

Frequency response limits the range over which a single effective height, impedance,
and pattern can be used. The antenna's electrical dimensions change with $\lambda$.
Its reactance changes, its feed match shifts, and a receiver tuned for one carrier can
reject nearby frequencies. A narrow response improves selectivity against unwanted
signals but can distort a wideband pulse or modulated waveform. A broad response passes
more spectrum but also admits more noise and interference. These are terminal and
system properties; the local far-zone field law still holds at each frequency component
within the linear range.

### Measurement validation sequence

Validate a dipole-reception measurement in a sequence that changes one physical
assumption at a time. First confirm the receiver frequency and reference level using a
known source or calibrated field. Then rotate the receiving rod through the local
electric-field direction. The response should have a maximum near parallel alignment
and a minimum near a right angle. Rotate a small loop separately to test its magnetic
flux response. A disagreement between the two orientation checks can expose a cable
pickup or a local near-field contribution.

Next change range at a fixed broadside geometry. Record voltage amplitude rather than
only a normalized detector scale. In a verified far field, the voltage from a fixed
electric probe should scale close to $1/r$, and a matched received-power reading should
scale close to $1/r^2$. A trend that changes abruptly with a small height adjustment
points toward reflection interference. A trend that remains unchanged with distance can
indicate receiver background or direct coupling in a shared instrument setup.

Finally, make the angular pattern scan at the chosen range. Revisit the broadside
maximum and an axial null between scan segments. Those repeated points measure source
drift and receiver drift. Save the raw voltage or power values with the exported polar
plot. Raw values retain calibration information and allow a new normalization if a
later reference measurement changes.

## Zone diagnostics and source constraints

The near, induction, and radiation terms can be compared with the dimensionless range
parameter

$$
kr=\frac{2\pi r}{\lambda}.
$$

A small harmonic dipole has electric-field components with the following distance-order
structure when evaluated at retarded time:

$$
E_r\sim\frac{p}{r^3}+\frac{\dot p}{cr^2},
\qquad
E_\theta\sim\sin\theta\left(
\frac{p}{r^3}+\frac{\dot p}{cr^2}+\frac{\ddot p}{c^2r}
\right).
$$

The symbols indicate scaling and omit convention-dependent signs and numerical factors.
The first term is associated with the dipole charge distribution. The second follows the
rate of change of dipole moment. The third follows the dipole acceleration and remains
as the radiating transverse field. A correct near-source calculation retains all terms
that are comparable at its selected value of $kr$.

For $kr\ll1$, the $1/r^3$ term dominates. A short electric probe sees strong radial and
axial structure. A loop can couple strongly to a nearby current even when the distant
radiation is weak. The electric and magnetic measurements can have a large phase
difference, so a plane-wave conversion from one field magnitude to the other is invalid.
For $kr\gg1$, the $1/r$ transverse term dominates and the field approaches a local plane
wave. The simple dipole pattern, $E=cB$, and radial power-flow interpretation apply in
that asymptotic region.

The transition is frequency dependent. At 1 MHz, one wavelength in free space is about
300 m. A point 10 m from a compact source has $kr\approx0.21$ and is deeply within the
reactive region. At 100 MHz, the same 10 m distance corresponds to $kr\approx20.9$,
which supports a far-zone treatment for a sufficiently small source in a low-reflection
environment. Distance in metres alone cannot classify an antenna measurement.

Harmonic phase also separates the field terms. If $p(t)=p_0\cos\omega t$, then
$\dot p(t)$ is a sine wave shifted by one quarter cycle and $\ddot p(t)$ is opposite in
sign to $p(t)$. A near-field probe may therefore show a large signal at a phase where a
far-zone receiver sees a different phase. Comparing raw phase traces at two distances is
an effective diagnostic, provided the reference oscillator, cables, and instrument
channels have been phase-calibrated.

The drive circuit constrains $p_0$ and $I_0$. A prescribed source voltage does not hold
both quantities fixed as frequency changes because antenna reactance, radiation
resistance, feedline transformation, and source impedance change the current. A
prescribed current source can hold $I_0$ more nearly constant over a limited band, but
it needs whatever terminal voltage the antenna impedance demands. State the controlled
source variable before applying a frequency scaling law or comparing two antennas.

Receiver placement also changes the source in the near zone. A conducting probe or a
large loop placed close to a driven dipole changes capacitance, inductance, and loss of
the source assembly. This back-action shifts the current distribution and can change the
very field being measured. Use a probe much smaller than the wavelength and source
geometry, retain sufficient separation, and verify that source current does not shift
when the probe is moved into position. Far-zone receiving antennas usually perturb the
source much less because their coupling is mediated by the outward radiation field.

For pulsed radiation, the source has a frequency spectrum rather than a single
wavelength. Each spectral component has its own $kr$, antenna pattern, and receiver
transfer function. A time-gated measurement can separate an early direct pulse from
later reflected pulses when the path difference exceeds the receiver time resolution.
The pulse amplitude still needs bandwidth calibration; a narrow receiver reshapes the
time trace and spreads a short pulse over a longer interval.

### Reporting a radiation measurement

A radiation result should identify the physical quantity at every stage. An electric
probe may report open-circuit voltage in volts. A calibrated field probe may report RMS
electric field in volts per metre. A spectrum analyzer may report power at a reference
impedance in dBm. A receiving antenna followed by a matched detector reports power
after its own effective aperture, losses, and polarization response. Converting one
quantity into another requires an antenna factor, impedance, gain, or calibration curve;
the displayed number alone does not contain that conversion.

Record the transmitter current or accepted power when possible. Source-generator power
can stay constant while feedline loss or mismatch changes accepted antenna power. In a
short-dipole comparison, a current monitor at the feed helps distinguish a true change in
radiation pattern from a changed source amplitude. In a receiving experiment, record
the receiver input impedance and whether the reported voltage is open-circuit, loaded,
or corrected to a reference plane.

Use an angular datum tied to the antenna axis. With a dipole, state whether the reported
angle starts at the axis or at broadside. A graph labeled only from zero to 180 degrees
can represent either convention and reverse the location of the expected null. State
the direction of propagation, the transmitter rod orientation, and the receiver rod or
loop orientation. These geometric entries are part of the measurement, not decorative
diagram labels.

The final report should separate three outcomes:

- **Model agreement.** Broadside maximum, axial null, $1/r$ field scaling, and the
  expected receiver orientation response agree within the stated uncertainty.
- **Random spread.** Repeated samples fluctuate around a stable mean because of detector
  noise, source drift, or positioning repeatability.
- **Systematic departure.** A raised null, displaced lobe, altered distance law, or
  phase ripple persists when a sample is repeated. Change geometry or instrumentation to
  identify the cause before assigning the feature to the antenna.

The record associates the short-dipole equations with a real source, propagation region,
receiving system, and uncertainty budget. It also supports comparison with a later
measurement at another frequency or range without silently changing the antenna model.