Did Heaviside Delete Maxwell?
Did Heaviside Delete Maxwell? Scalar Waves and Longitudinal Fields in Radio Science
Oliver Heaviside replaced Maxwell’s sprawling component notation with the compact field equations radio amateurs know today. Did his simplification merely improve the map—or did it erase real electromagnetic territory?
I started getting questions from several radio amateurs who follow Hans G. Schantz: Who was Oliver Heaviside? Did he really replace Maxwell’s original equations? And did the modern formulation leave behind scalar or longitudinal electromagnetic waves? The short answer is that Heaviside radically improved the notation, but he did not remove a verified physical degree of freedom. The longer answer is more interesting—especially for anyone who works with antenna near fields.
The Answer Before the History
There is a genuine historical change underneath the provocative title. Maxwell gave the scalar and vector potentials a prominent place in his presentation. Heaviside preferred to work directly with electric and magnetic fields and intentionally pushed the potentials into the background.
That was a change of variables and emphasis. It was not the discovery that most of Maxwell’s experimentally supported solutions were wrong, nor did it create an opening for a hidden family of lossless “scalar” radio waves.
Who Was Oliver Heaviside?
Oliver Heaviside (1850–1925) was an English telegraph engineer and largely self-taught mathematical physicist. He began his career working with telegraph circuits, where distortion, delay and loss were practical engineering problems rather than classroom exercises.
His transmission-line equations connected resistance, inductance, capacitance and leakage to signal propagation and distortion.
He replaced awkward component and quaternion notation with divergence, curl and compact vector equations.
He treated electric and magnetic fields as the physically meaningful quantities and potentials as optional calculating tools.
For radio amateurs, Heaviside is not an obscure footnote. Every discussion of transmission lines, loading, distortionless propagation, impedance and modern Maxwell notation carries some of his fingerprints.
Did Maxwell Really Have Twenty Equations?
Yes—but the number needs context. In Maxwell’s 1865 presentation and later summaries, three-dimensional vector relations were commonly written as separate equations for the x, y and z components. The list also mixed fundamental field laws with definitions, constitutive relations and continuity equations.
Counting each Cartesian component separately can give twenty scalar equations. Writing one three-component relation as a single vector equation immediately reduces that count. Moving Ohm’s law, material relations and definitions into their own chapters reduces it again.
| Maxwell’s broader list included | Where it lives in modern notation | Was it lost? |
|---|---|---|
| Electric charge and displacement | ∇·D = ρ |
No |
| Magnetic induction and its vector potential |
∇·B = 0 and B = ∇×A
|
No |
| Electromagnetic induction | ∇×E = −∂B/∂t |
No |
| Conduction plus displacement current | ∇×H = J + ∂D/∂t |
No |
| Charge conservation | ∇·J + ∂ρ/∂t = 0 |
No; it follows consistently from the field equations |
| Electric conductivity | J = σE |
No; it is treated as a material relation |
| Electric and magnetic material response |
D = εE and B = μH for simple media |
No; these remain constitutive relations |
The bookkeeping trap: comparing “twenty scalar component equations” with “four vector equations” is not a comparison of twenty physical effects with four physical effects. One modern vector equation can contain three scalar component equations, and several of Maxwell’s other relations are still used alongside the famous four.
What About Maxwell’s Quaternions?
Another popular story says that Maxwell’s full quaternion algebra contained an extra scalar component and that Heaviside deleted it. This confuses an algebraic product with a physical field.
For two vectors, quaternion multiplication packages a scalar-product-like part and a vector-product-like part together. Modern vector analysis writes those operations separately as a dot product and a cross product. Splitting the package does not throw either operation away.
Quaternion-style packaging: one product can contain both a scalar and a vector part.
Modern vector analysis: the same useful operations appear separately as ∇·A and ∇×A.
A scalar produced by the mathematics is not automatically a new measurable scalar field with its own energy channel.
Maxwell used quaternion notation mainly to summarise and interpret results already developed in components. He explicitly selected the scalar or vector part appropriate to the physical relation. There is no historical basis for treating every unused part of every quaternion product as a suppressed mode of nature.
What Heaviside Actually Removed: Potentials from the Foreground
The historically real “deletion” concerns the scalar potential φ and vector potential A. Maxwell considered the vector potential important enough to call it electromagnetic momentum. Heaviside regarded the fields as clearer and more physical, so he formulated the theory directly in terms of E, D, B and H.
The potentials generate the familiar fields:
B = ∇×A
E = −∇φ − ∂A/∂t
If we calculate with the fields, the potentials may disappear from the main equations. If we calculate with potentials, the same electric and magnetic fields reappear after differentiation. Classical predictions do not depend on which description we prefer.
Why Potentials Are Not Unique: Gauge Freedom
An electric potential is like height on a map. We may call sea level zero metres or choose another reference; slopes between locations remain the same. The electric field depends on that slope, not on the arbitrary zero.
Time-varying electromagnetism has a more general version of the same freedom. For a suitably smooth function χ, we may make the transformation:
A′ = A + ∇χ
φ′ = φ − ∂χ/∂t
The resulting E and B fields are unchanged.
This means that a particular numerical value of A at one point cannot by itself be an observable. Different gauges assign different values while predicting the same measurement.
Does the Aharonov–Bohm Effect Make the Potential “Real”?
Quantum mechanics makes the story more subtle. In the Aharonov–Bohm experiment, electrons travel through a region where the local electric and magnetic fields can be zero, yet the interference pattern depends on magnetic flux enclosed by the two paths.
The measurable phase difference is proportional to:
Δθ = (q/ℏ) ∮ A·dl = (q/ℏ) ΦB
This proves that local E and B values along each electron path are not the whole quantum story. The closed-loop phase—or holonomy—is gauge invariant and measurable.
It does not prove that one chosen, gauge-dependent value of A is directly observable. Saying “the vector potential is physically real” is therefore an interpretation of the experiment, not the only conclusion forced by it.
Four Different Things Called a “Scalar” or “Longitudinal Wave”
Much of the confusion disappears when we stop using one label for four different ideas.
| Phrase | What it can legitimately mean | What it does not establish |
|---|---|---|
| Scalar potential | A scalar function φ used with A to calculate E and B. |
That φ is a separately observable radiation field. |
| Scalar representation | One or more scalar functions used mathematically to generate ordinary electromagnetic solutions. | That the resulting radiation has scalar polarization or carries extra energy. |
| Longitudinal field component | A component of E or H parallel to a chosen direction, common in near fields, waveguides and bounded structures. |
That a new free-space radiation mode exists. |
| Independent longitudinal vacuum wave | A proposed propagating mode with its field parallel to its wave vector. | Standard source-free Maxwell theory does not supply such a mode. |
Why Ordinary Free-Space Radiation Is Transverse
Consider a small piece of a source-free electromagnetic wave. Write its spatial and time dependence as ej(k·r−ωt). In an empty homogeneous region, Gauss’s laws say:
∇·E = 0 and ∇·B = 0
For a plane-wave component, these become k·E = 0 and k·B = 0.
In plain language: neither field points along the propagation vector k. Both are transverse.
This is not an arbitrary convention imposed by Heaviside. It follows directly from the source-free field equations. A new independent longitudinal radiation mode would require different sources, a material medium, boundaries, or modified field equations—and therefore a new, testable prediction.
But Antennas Really Do Have Longitudinal Near Fields
Here Schantz is pointing to something real. The exact field of an oscillating electric dipole is not purely transverse at every distance. Close to the antenna, the electric field has a radial component. Depending on the term, the field strength falls approximately as 1/r³, 1/r² or 1/r.
| Field term | Approximate fall-off | Engineering interpretation |
|---|---|---|
| Quasi-static term | 1/r³ |
Strongly stored electric or magnetic energy close to the antenna. |
| Induction term | 1/r² |
Near-field coupling, including radial or longitudinal components. |
| Radiation term | 1/r |
The field that survives at long range and carries finite power through expanding spheres. |
A radial near-field component is “longitudinal” relative to the radius from the antenna. It can couple power into a nearby antenna, just as two transformer windings can couple strongly while arranged along the same axis. Near-field communication and inductive links are entirely real.
But that radial component is not an additional free-space polarization that survives into the far field. Its amplitude falls too rapidly. For example, a 1/r² field has an energy density scaling roughly as 1/r⁴; it cannot maintain a finite radiated-power channel through spheres whose area grows as r².
Does the Far-Field Approximation Violate Gauss’s Law?
Schantz notes that if we retain only the familiar dipole radiation term—an electric field proportional to sin θ/r—and then treat it as an exact all-distance solution, its divergence leaves a smaller term of order 1/r². The exact dipole solution includes a radial 1/r² field that cancels that remainder.
The calculation is useful. The dramatic wording is not.
An asymptotic approximation is not expected to reproduce discarded orders exactly. The far-field expression keeps the leading 1/r term and neglects faster-decaying terms. Finding an error at the neglected 1/r² order does not reveal a failure of textbook electromagnetism; it tells us where the approximation stops being exact.
For a ham, the practical analogy is familiar: a transmission line may be treated as lossless when attenuation is irrelevant to the calculation. That model is not “violating conservation of energy” because real coax has loss. It is a controlled approximation with a stated range of usefulness.
What Schantz Gets Right—and Where an Eyebrow May Rise
| Claim or theme | Assessment |
|---|---|
| Heaviside did not discard sixteen independent physical laws. | Correct. The numerical comparison mostly reflects notation, grouping and the separation of constitutive relations. |
| Heaviside deliberately removed potentials from the primary formulation. | Historically fair. He preferred field quantities, but this did not reduce the classical solution space. |
| Scalar mathematical representations do not imply exotic scalar radiation. | Correct. A scalar function can encode an ordinary transverse field. |
| Longitudinal or radial electric components occur around antennas. | Correct. They are part of the exact conventional near-field solution. |
| Those components are “real, honest-to-goodness longitudinal waves.” | Terminologically risky. They are retarded, time-varying fields, but not an independent propagating vacuum mode. |
| The Aharonov–Bohm effect proves the vector potential itself is real. | Too strong. The experiment proves a gauge-invariant loop phase and enclosed flux matter; the ontology of a particular gauge potential remains interpretive. |
What This Means at the Ham Station
- Friis is a far-field equation. It should not be used to describe close transformer-like or capacitive coupling.
- A dipole null is a far-field statement. Two nearby antennas can couple strongly in an orientation that appears to place one in the other’s far-field null.
- Near-field E and H need not have the 377 Ω plane-wave ratio. A small loop and an E-field probe can therefore respond very differently close to the same source.
- A strong local field does not prove good radiation efficiency. It may indicate stored reactive energy or coupling to nearby conductors.
- Calling a link “scalar” adds no engineering content. A useful claim must predict a measurable result that ordinary Maxwell modelling does not.
How to Evaluate an Extraordinary “Scalar-Wave” Claim
Specify received voltage, field component, phase, power transfer or force—not only a potential or mathematical scalar.
Compare the claim with an exact Maxwell solution or full-wave simulation including the real source, conductors and return paths.
State a distance law, polarization, shielding response or phase behaviour that conventional near-field coupling cannot reproduce.
- Measure at several distances. A near-field term and a radiation term have different fall-off rates.
- Control ordinary capacitive, inductive, common-mode and conducted paths.
- Change orientation and shielding while keeping source power and geometry documented.
- Measure accepted and received power with calibrated reference planes.
- Ask whether the proposal conserves energy and remains gauge invariant.
If a claimed scalar link produces exactly the range, orientation and distance behaviour predicted by conventional electric or magnetic near-field coupling, the new label has not identified new physics.
Primary and Source Reading
- Hans G. Schantz — recorded talk and transcript: “Did Heaviside Delete Maxwell?”
- Hans G. Schantz — talk abstract
- James Clerk Maxwell — General Equations of the Electromagnetic Field
- The Feynman Lectures — exact fields of an oscillating dipole
- Y. Aharonov and D. Bohm — Significance of Electromagnetic Potentials in the Quantum Theory
- A. Tonomura et al. — shielded-flux Aharonov–Bohm experiment
Conclusion
Heaviside did replace Maxwell’s presentation, in the sense that the compact vector form used today is largely his reformulation. He did not replace Maxwell’s successful physics with an incomplete theory.
- The move from twenty scalar equations to four vector equations was mainly a change in notation and organisation.
- Potentials were removed from the foreground, not prohibited or made mathematically unavailable.
- Quaternion algebra did not contain a hidden physical scalar mode waiting to be restored.
- Radial and longitudinal near-field components are real and important to antenna coupling.
- Those components are not an extra long-range vacuum radiation channel.
- The Aharonov–Bohm effect reveals global quantum structure, but it does not make a gauge-dependent local potential directly observable.
Mini-FAQ
- Did Heaviside invent Maxwell’s equations? He reformulated and organised Maxwell’s theory into much of the compact vector form used today. The physical unification and displacement-current insight came from Maxwell.
- Were sixteen equations deleted? No. Component equations were bundled into vectors, while definitions and material relations were moved outside the famous set of four.
- Are potentials merely fake mathematics? They are gauge-dependent, but extremely useful and central to modern quantum theory. Gauge-invariant combinations of them have measurable consequences.
- Do longitudinal electric fields exist? Yes—in near fields, bounded structures, plasmas and guided systems. That is different from an independent longitudinal radiation mode in source-free vacuum.
- Can near-field coupling carry useful power? Absolutely. Transformers, NFC, wireless charging and closely coupled antennas rely on it. The power transfer does not require a new scalar wave.
- Is Schantz’s talk fringe physics? Its central answer rejects the strongest fringe claim. The caution is that some of its terminology makes ordinary near-field physics sound more revolutionary than it is.
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