Dipole 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 $D ≪ λ$ , 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 $E = c B$ 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 $E = c B$ in magnitude, and have a radial time-averaged Poynting vector. The energy-flow and radiation-pressure relations belong to Electromagnetic Momentum. The present lesson uses only the local radiation-zone intensity needed for antenna patterns and receiver estimates.

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.

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

p (t) = p (t) \overset{z}{^} .

A pair of charges $+ q$ and $- q$ separated by vector $ℓ$ has moment $p = q ℓ$ . 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.

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.

For harmonic drive, use

p (t) = p_{0} cos ω t, ω = 2 π f, λ = \frac{c}{f} .

The emitted radiation has the drive frequency $f$ and the corresponding free-space wavelength $λ$ . 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 $ω^{2} p_{0}$ . 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 $θ$ from the dipole axis. The leading radiation-zone electric field has the following magnitude when source size is small compared with both $r$ and $λ$ :

∣ E_{θ} (r, θ, t) ∣ = \frac{1}{4 π ε _{0} c ^{2}} \frac{∣ p ¨ ( t - r / c ) ∣}{r} sin θ .

The corresponding magnetic field is azimuthal and has magnitude

∣ B_{ϕ} ∣ = \frac{∣ E _{θ} ∣}{c} .

The expression identifies four independent controls. Field magnitude decreases as $1/ r$ . It vanishes on the dipole axis because $sin θ = 0$ . 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.

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.

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_{θ, rms} = \frac{p _{0} ω ^{2}}{4 π ε _{0} c ^{2} r 2} sin θ .

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

⟨ S_{r} ⟩ = \frac{E _{θ, rms}^{2}}{μ _{0} c} = \frac{p _{0}^{2} ω ^{4}}{32 π ^{2} ε _{0} c ^{3} r ^{2}} sin^{2} θ .

The angular dependence is the characteristic electric-dipole pattern. The power per unit solid angle is proportional to $sin^{2} θ$ . 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.

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.

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.

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.

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

\frac{d P}{d Ω} = r^{2} ⟨ S_{r} ⟩ = \frac{p _{0}^{2} ω ^{4}}{32 π ^{2} ε _{0} c ^{3}} sin^{2} θ .

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

0 \int 2 π 0 \int π sin^{2} θ sin θ d θ d ϕ = \frac{8 π}{3} .

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

P_{rad} = \frac{p _{0}^{2} ω ^{4}}{12 π ε _{0} c ^{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 $ω^{4}$ factor in $P_{rad}$ 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 $I_{0} / ω$ , so the dipole-moment amplitude is

p_{0} \approx \frac{I _{0} ℓ}{ω}, ℓ ≪ λ .

Substitution gives a power proportional to $I_{0}^{2} ℓ^{2} ω^{2}$ . With RMS current, the standard short-dipole radiation resistance is

R_{rad} = 80 π^{2} (\frac{ℓ}{λ})^{2}, P_{rad} = I_{rms}^{2} R_{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 $R_{rad}$ rather than those loss resistances.

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.

The formula is restricted to $ℓ ≪ λ$ . 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 $ℓ / λ$ , 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 $R_{rad} \propto (ℓ / λ)^{2} \propto f^{2}$ . 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 $p_{0} ω^{2} / r$ . 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_{oc, rms} = h_{eff} \cdot E_{rms} = h_{eff} E_{rms} cos α .

The angle $α$ is between the antenna effective-height vector and the incoming electric-field vector. The scalar $h_{eff}$ 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 $cos α$ . If the receiver is connected to a resistive matched input and all other conditions are fixed, the received power varies as $cos^{2} α$ . 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.

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.

Antenna terminal model

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

Z_{A} = R_{rad} + R_{loss} + j X_{A} .

$R_{rad}$ represents reradiation from current in the receive antenna. $R_{loss}$ 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

V_{L} = V_{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.

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.

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_{avail} = \frac{V _{oc, rms}^{2}}{4 R _{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 $\overset{n}{^}$ , a spatially uniform magnetic field produces flux $N B A cos β$ , where $β$ is the angle between the field and the loop normal. An electrically small loop in a sinusoidal plane wave has RMS induced emf

E_{rms} = N A ω B_{rms} ∣ cos β ∣ = N A \frac{2 π f}{c} E_{rms} ∣ cos β ∣.

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.

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.

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} (θ) = \frac{⟨ S _{r} ( θ )⟩}{⟨ S _{r} ( π /2 )⟩} = sin^{2} θ .

The corresponding normalized electric-field amplitude pattern is $F_{E} (θ) = ∣ sin θ ∣$ . 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.

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.

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} ≳ \frac{2 D ^{2}}{λ} .

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 $r ≫ λ$ statement.

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.

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} (θ) = \frac{P _{R} ( θ )}{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{δ P _{R}}{P _{R}} \approx 2 \frac{δ r}{r} .

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

u_{rel} = u_{cal}^{2} + u_{range}^{2} + u_{angle}^{2} + u_{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

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.

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 $E = c B$ , $1/ r$ field scaling, or an angular far-field pattern.
Angle check. Measure polar angle from the dipole axis. Broadside is $θ = 9 0^{\circ}$ , not $θ = 0$ .
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 $sin^{2} θ$ 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_{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 λ ^{2}}{4 π},

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 $S A_{e}$ .

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 $λ$ . 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

k r = \frac{2 π r}{λ} .

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{p ˙}{c r ^{2}}, E_{θ} \sim sin θ (\frac{p}{r ^{3}} + \frac{p ˙}{c r ^{2}} + \frac{p ¨}{c ^{2} r}) .

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 $k r$ .

For $k r ≪ 1$ , 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 $k r ≫ 1$ , the $1/ r$ transverse term dominates and the field approaches a local plane wave. The simple dipole pattern, $E = c B$ , 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 $k r \approx 0.21$ and is deeply within the reactive region. At 100 MHz, the same 10 m distance corresponds to $k r \approx 20.9$ , 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 $p (t) = p_{0} cos ω t$ , then $\overset{p}{˙} (t)$ is a sine wave shifted by one quarter cycle and $\overset{p}{¨} (t)$ 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 $k r$ , 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.