Open almost any treatment of retarded potentials and you will find, somewhere among the worked examples, a sphere being charged. The potential outside it is written down in one line:
the applied voltage, scaled by distance, delayed by the light travel time. It is the natural first exercise in retardation, and it looks entirely innocent.
There is a question that exercise does not ask, and the answer to it turns out to govern everything else: where is the charge coming from?
Continuity does not negotiate
Start with the conservation of charge, which is not a modelling assumption but a structural feature of electromagnetism:
Integrate it over any volume containing the sphere. The rate of change of the enclosed charge equals minus the current flowing out through the bounding surface. Rearranged: if the enclosed charge is changing, current must be crossing the surface. There is no alternative — not an approximation, not a limit, an identity.
Now impose exact spherical symmetry, which is what makes the potential above take its simple form. A current crossing the surface has a direction. A direction breaks the symmetry.
So the two requirements are incompatible. A perfectly spherically symmetric source whose charge varies in time cannot exist. Not “is difficult to build” — cannot exist, because it would require charge to appear where none was, and nothing in the theory permits that.
This is worth pausing on, because the object in question is not exotic. A time-varying monopole is what one writes down without thinking whenever a charging sphere is treated as spherically symmetric, and it appears in that role in a great deal of otherwise careful work.
Writing it down anyway
Impossible objects can still be written down, and doing so is often instructive: the formalism, asked to describe something that cannot be, tends to say so. What is interesting here is how many different ways it finds to say it.
Take the exterior region seriously as a boundary-value problem. Prescribe the potential above on a spherical boundary, ask for a solution of the wave equation outside it, and impose the Lorenz gauge condition. This is a well-posed mathematical question, whatever its physical status.
The answer is startling. Outside the boundary, while the applied voltage is still changing, the gauge-invariant electric field is exactly zero.
Not small. Zero, identically, at every radius. The magnetic field vanishes by symmetry, and the electric field vanishes because the two terms that make it up cancel term by term:
The gradient of the scalar potential is not zero — it is the ordinary Coulomb-like field one would expect, plus a term proportional to the rate of change of the voltage. But the physical field is that gradient plus the time derivative of the vector potential, and the vector potential required by the Lorenz condition for this source is precisely the one whose time derivative cancels it.
This is not an approximation valid at large distance, or at low frequency, or to leading order. It is an identity, and it has been checked four separate ways: by direct algebraic construction, by the classical equation of motion including the canonical momentum, by the Aharonov–Bohm phase around a closed loop, and by the quantum multipole selection rule, which forbids a monopolar source from emitting a real photon at all because photon emission requires angular momentum a monopole does not have.
A fifth check, using Gauss’s law in the exterior vacuum, is sometimes offered as a fifth proof. It is not one, and it is worth being clear about why. If the field is identically zero then its divergence is zero by construction, so the calculation cannot fail. It verifies that the construction is not internally inconsistent, which is worth knowing, but it does not distinguish a vanishing field from any other divergence-free field in the exterior — including the ordinary Coulomb field.
What the configuration is not
Here is where the impossibility resurfaces, and where the result has to be stated with more care than it first appears to need.
If the exterior field is exactly zero, then the flux of that field through any surface enclosing the boundary is also zero. Gauss’s law in integral form then says the enclosed charge is zero.
But the potential we started from is monopolar, and the coefficient in it is usually read as a charge: the sphere charged to voltage V holds a charge proportional to V. The two statements cannot both be about the same quantity.
The resolution is that the coefficient is a boundary parameter of the retarded solution, not a demonstrated physical charge sitting anywhere. The exterior problem was posed by prescribing a potential on a boundary; it says nothing about the interior, and establishing that the coefficient corresponds to a real surface charge would require solving the interior problem and matching across the boundary. That has not been done, and it is not a formality: for a genuine surface charge the field must be discontinuous across the surface, so a vanishing exterior field would require a non-zero interior field, which a conductor cannot have.
So the configuration with the exactly vanishing exterior field is not a charged conductor. It is an exterior boundary-value problem that carries no net enclosed charge, and the informal phrase “a sphere charged to potential V” should not be read as asserting otherwise.
Three symptoms, one cause
Once this is seen, several separate difficulties turn out to be the same difficulty.
The vector potential constructed above contains the time integral of the applied voltage. If the source settles to a constant voltage, that integral grows without bound, so the constructed vector potential diverges at fixed radius even though the potential itself stays finite. A vector potential growing without limit in a system that has physically reached electrostatic equilibrium is a signal that the solution has stopped being the physical one — it is no longer anchored to a current history, because there is no longer a current.
The flux does not balance, as above
And most directly: applied to a source fed through a single point with no return path included, the Lorenz gauge condition itself is violated substantially. Not marginally — the numbers do not come close. Including an explicit return conductor restores it to within a tenth of a per cent, verified across several dozen field points spanning two decades in distance and the full angular range.
These are not three unrelated problems. They are the impossibility asserting itself at three different places in the formalism: a source whose charge varies without a path for the charge to arrive along is not a solution of Maxwell’s equations, and the machinery says so wherever one happens to look.
There is a moral here about gauge conditions that is easy to miss. The Lorenz condition is usually presented as a convenience, a choice made to decouple the potential equations. It is more than that. Substituted into the field equations it enforces charge conservation, and a source that does not conserve charge cannot satisfy it. What looks like a free choice of bookkeeping is a physical requirement wearing a bookkeeper’s coat.
What the real configuration gives
With the return path included, the comparison can finally be made honestly, and the outcome has two halves that behave quite differently.
The field cancellation degrades, but not by much. The gauge-invariant field is now small rather than exactly zero, tilted away from purely radial by an amount varying systematically with angle relative to the injection point, and the magnetic field is no longer zero. The physical energy flux, identically zero in the idealised problem, becomes non-zero: around 1.7% of the energy stored in the circuit.
Small, genuinely not zero, and it would be wrong to round it down. That figure deserves one qualification. It is a measurement made at a particular place and for a particular transient — five sphere radii out, with the voltage rising over about ten nanoseconds — and it is not a constant of the configuration. Measured further out it grows; measured for a slower transient it shrinks. But the two are not independent: recomputing the geometry by an independent route shows that both collapse onto the single ratio between the size of the system and the distance light travels while the source is rising. Hold that ratio fixed and the deviation holds fixed too, across a factor of sixteen in observation radius.
Which is the more useful statement, because it says what the deviation is measuring. Not a residual inaccuracy, and not a small effect that happens to be there — it is the distance from the quasi-static, written as a ratio of lengths. For a sphere a few centimetres across charged over ten nanoseconds, light crosses the geometry some hundred times during the rise, and a per cent or two is what that buys.
The kinematic identity does not degrade at all. The scalar quantity built from the wavefront of the potential is computable exactly from the behaviour of that potential alone, and the potential of a charged sphere does not depend on how the charge got there. That identity holds unchanged between the idealised and the real configuration.
So two claims that looked equally strong are not equally robust, and the difference is structural rather than a matter of degree. One depends on the currents, and the currents are exactly what a real feed point disturbs. The other depends only on the charge, and a monopole is a monopole however it was assembled.
Why this matters beyond the sphere
The general statement is short. Continuity is not negotiable; what can be chosen is only what carries it.
That reading makes several apparently unrelated situations look the same. A bar of metal polarised by an external field has conduction current that stops at its ends — and the circuit closes through displacement current in the space beyond, which is why the bar radiates while it settles rather than simply arriving at equilibrium. In a high-altitude nuclear detonation, the Compton current streams outward with no return path at all, and the circuit closes because the same electrons ionise the air behind them into a conductor: the source manufactures its own return path while acting, and that process is what limits the pulse.
Three configurations, one rule. In each case the interesting physics is not in the outgoing current, which is obvious, but in how the path closes, which is not.
And it suggests a habit worth keeping. When a calculation involving a time-varying source produces something strange — a quantity growing without bound, a flux that does not balance, a gauge condition that will not hold — the first thing to check is not the algebra. It is whether the charge has somewhere to come from.
Sources
S. Filippi, “Field Exclusion for a Monopolar Source in Lorenz Gauge,” 2026.
S. Filippi, “Dynamic Open-Path Sources and the Lorenz Gauge: Resolving a Charging Sphere with Single-Point Injection,” 2026.
S. Filippi, “The Scalar Longitudinal b-Channel in Lorenz Gauge: Foundations and an Explicit Coulomb-Gauge Verification,” 2026.
C. Galeriu, “The Lorenz gauge condition and charge conservation,” 2022.
J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1998, Ch. 6.
Y. Aharonov and D. Bohm, “Significance of Electromagnetic Potentials in the Quantum Theory,” Phys. Rev., vol. 115, pp. 485-491, 1959.
Preprints and technical notes are deposited on Zenodo under ORCID 0009-0007-3660-8498.
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