Jacob's Ladder: Why the Spark Climbs

The apparatus: a V of wire and a current-limited transformer

A Jacob's ladder is two stiff conductors arranged as a narrow-bottomed V — the tips are a few millimetres apart at the bottom and diverge to several centimetres at the top. They are driven by a high-voltage, current-limited source, most commonly a neon-sign transformer delivering roughly 9–15 kV at a short-circuit current of only 30–60 mA. Oil-burner ignition transformers (~10 kV) and, in more dangerous home builds, microwave-oven transformers are also used.

The word 'current-limited' is not incidental — it is what makes the device work at all. An established electric arc has a negative differential resistance: as more current flows, the channel heats, ionizes further, and its voltage drops. An ideal voltage source connected to such a load would run away to a destructive short. A neon-sign transformer instead has a large deliberate leakage reactance (magnetic shunts between primary and secondary) that behaves like a series ballast, holding the current near a fixed value regardless of the arc's shifting resistance. This ballast both stabilizes the arc and caps the fault current — the same reason fluorescent tubes and arc lamps need a series inductor or resistor.

Why the strike is always at the bottom: Paschen's law

Air is an excellent insulator until the field across it reaches its dielectric strength, about E₀ ≈ 3 MV/m (30 kV/cm) for centimetre gaps at standard temperature and pressure. A stray free electron — there are always some, from cosmic rays and background radioactivity — is then accelerated hard enough between collisions to ionize a nitrogen or oxygen molecule (ionization energies ≈ 15.6 eV and 12.1 eV). Each new electron does the same, and the population grows exponentially: n(x) = n₀·e^{αx}, the Townsend avalanche, with α the first Townsend ionization coefficient. When the avalanche and its photoionization feedback bridge the gap, a conducting channel — a spark — forms.

The voltage needed to do this depends on the gap through Paschen's law (Friedrich Paschen, 1889):

• V_b = B·p·d / [ln(A·p·d) − ln(ln(1 + 1/γ))]

where p is pressure, d the gap, γ the secondary-emission coefficient, and A, B gas constants. The breakdown voltage as a function of the product p·d has a minimum — for air about 327 V at p·d ≈ 0.57 Torr·cm — and rises to either side. For the millimetre-to-centimetre gaps of a ladder at atmospheric pressure we are far up the right-hand branch, where V_b grows roughly linearly with gap d (a few kV per mm).

That single fact explains the whole starting behavior. With the transformer holding a fixed peak voltage, breakdown occurs wherever the local V_b(d) first falls below the applied voltage — and since V_b increases with the gap, the smallest gap has the smallest V_b. The bottom of the V, where the wires nearly touch, is therefore where the air gives way first. A 12–15 kV supply breaks down a gap of roughly 4–6 mm, which is exactly the bottom spacing a builder chooses.

From spark to plasma: the thermal arc

The initial spark is a thin, cold, field-driven filament, but it dumps energy into the channel fast. Ohmic heating (P = I²R per unit length, or σE² per unit volume) raises the gas temperature until ionization becomes thermal rather than field-driven: at several thousand kelvin, ordinary collisions in the hot gas are energetic enough to keep the plasma ionized. The channel transitions from spark to a self-sustaining electric arc — a thermal plasma whose core runs at roughly 6,000–20,000 K.

The degree of ionization in local thermodynamic equilibrium is set by the Saha equation, n_e·n_i/n_0 ∝ T^{3/2}·e^{−E_ion/kT}: even a few percent ionization gives the channel a conductivity high enough to carry the current at a modest field. Radially, the arc column settles into the balance described by the Elenbaas–Heller equation — ohmic input σ(T)E² is carried outward by thermal conduction, σ(T)E² = −(1/r)·d/dr[r·κ(T)·dT/dr] — which fixes the arc's temperature profile and radius. The upshot is the arc's signature: it needs a high voltage to strike but only a low voltage to burn, typically a column gradient of ~10–50 V/cm plus a combined cathode-and-anode fall of order ~15 V. Once lit, the ladder's arc is a hot, bright, low-voltage conductor — and that is the thing that now begins to move.

Why it climbs: buoyant convection (with a magnetic nudge)

The arc heats a slug of air to thousands of kelvin. At roughly constant atmospheric pressure the ideal-gas law makes density fall in inverse proportion to temperature, ρ ∝ 1/T, so the plasma channel is dramatically lighter than the room air around it. The buoyant (Archimedes) acceleration of the hot gas is

• a = g·(ρ_amb − ρ_hot)/ρ_hot = g·(T_hot/T_amb − 1)

For a 6,000 K core in 300 K air that is a ≈ 19·g — an upward push of order tens of g. Real rise speed is throttled by aerodynamic drag and by cool air entrained into the plume, so the visible arc climbs a 20–30 cm ladder in about one to three seconds (order 10–30 cm/s), not at rocket speed. But the direction is unambiguous: the hot channel floats, and because its two ends are anchored to the electrodes, the arc bows upward and its attachment points (the arc 'roots') are dragged along the diverging wires.

Convection is the dominant driver at the low currents of a demonstration ladder, and the cleanest proof is orientation: the ladder only works narrow-bottom, wide-top, in normal gravity. Invert it, lay it flat, or fly it in free fall and the climb disappears because buoyancy is what powers it. There is a second, smaller contribution at higher currents: the current runs up one leg, across the arc, and down the other, forming a loop, and the J × B (Lorentz) self-force on that loop pushes to enlarge it — the same 'motor' or railgun effect that blows arcs outward in circuit breakers. In a low-current ladder this magnetic push is a minor assist to convection; in a high-current arc it can dominate.

Why it snaps out at the top: arc lengthening and Ayrton's equation

As the arc rides up the widening V, its length keeps increasing — and a longer arc costs more voltage. Hertha Ayrton, whose meticulous measurements of the electric arc earned her a Royal Society paper and the book The Electric Arc (1902), captured this in the empirical Ayrton equation:

• V = a + b·ℓ + (c + d·ℓ)/I

where ℓ is the arc length and I the current. The essential term is b·ℓ: arc voltage rises roughly linearly with length. At the ladder's top, an arc stretched to 20–30 cm at, say, ~40 V/cm demands on the order of a kilovolt across the column alone. Two things then conspire. First, the required voltage climbs while the current-limited transformer's terminal voltage sags under load — most of the open-circuit kilovolts are dropped across the internal leakage reactance, leaving only ~1 kV for the arc — so the source can no longer push its current through the ever-longer column. Second, the stretched, thinning channel loses proportionally more heat to the surrounding air and to entrained cool gas, so it cools and its conductivity falls. When the voltage the current-limited supply can deliver can no longer sustain the lengthening column, the power balance tips negative, the channel cools below the ionization threshold, and the arc extinguishes — it 'snaps out' at the top.

On AC, the arc also passes through zero current twice per cycle, momentarily de-ionizing; near the top, where the arc is long and marginal, one of these zero-crossings is enough to finish it off. This is precisely the principle that circuit breakers exploit — arc chutes and magnetic blowout coils deliberately stretch and cool a fault arc until it can no longer restrike.

The rising buzz and the endless restrike

The instant the top arc dies, the gap is open again and the full transformer voltage reappears across the electrodes. The lowest breakdown voltage is once more at the bottom, so a fresh spark strikes there and the whole climb repeats — a self-restarting relaxation oscillator whose period is set by how fast the plume can loft the arc up the ladder.

The characteristic buzz is acoustic. On a 60 Hz supply the arc current, and hence its ohmic heating, peaks in magnitude twice per cycle, so the channel thermally expands and contracts at 120 Hz (100 Hz on 50 Hz mains), radiating a hum at that frequency plus a spray of higher harmonics from the turbulent, restriking discharge. As the arc lengthens on each climb the sound shifts and roughens, giving the familiar 'rising' quality, and each restrike at the bottom punctuates the loop with a sharp snap. What you hear is a direct readout of the physics: dielectric breakdown at the bottom, a plasma heating and lofting, and a stretched arc quenching at the top, over and over.

History, look-alikes, and where the same physics matters

The Jacob's ladder is a laboratory toy with a serious pedigree. Its climbing arc is a scaled, tamed version of the discharges studied by the pioneers of gas conduction — Paschen on breakdown voltage, John Townsend on the avalanche, and Ayrton on the arc's voltage-length-current law. Kenneth Strickfaden's spectacular electrical props for Frankenstein (1931) cemented it in popular imagination as the emblem of the high-voltage lab.

It is easy to confuse with its relatives, but the distinctions are sharp:

• Corona discharge is a partial, non-thermal breakdown confined to the thin, high-field shell around a sharp electrode; it never bridges the gap and stays near room temperature. The ladder's arc is a complete, thermal channel.
• Lightning is the same family of dielectric breakdown at kilometre scale, with leaders and a return stroke of tens of kiloamps — a transient thermal arc rather than a steadily climbing one.
• Tesla-coil and plasma-globe streamers are high-frequency RF discharges; the ladder runs at line frequency and depends on ordinary buoyant convection.

The controlled, high-current cousins of the ladder's arc are the workhorses of heavy industry: electric-arc furnaces melt scrap steel at ~10,000 K, arc welding exploits the same low-voltage thermal channel to fuse metal, and circuit breakers live or die by the opposite goal — lengthening, cooling, and blowing out an arc so it cannot restrike, exactly the failure mode that ends each climb. Open questions in arc physics remain surprisingly current: the detailed turbulence of the buoyant plume, non-equilibrium departures from the Saha and Elenbaas–Heller pictures at the arc's edges, and cathode-spot dynamics are all active research topics in plasma engineering.

The two regimes of one climb cycle: the cold strike versus the hot, rising arc

Property	The strike (dielectric breakdown)	The climb (sustained thermal arc)
Where / when	At the narrowest gap, the bottom, at first	Everywhere from bottom to top, over ~1–3 s
What limits it	Paschen breakdown voltage V_b(d)	Arc voltage rising with length (Ayrton)
Ionization mechanism	Field-driven Townsend/streamer avalanche	Thermal (collisional) ionization, Saha balance
Gas temperature	Near ambient until the channel forms	Thousands to ~20,000 K (hot core)
Voltage	High: full transformer voltage (kV)	Low: ~10–50 V/cm column + ~15 V electrode fall
What drives the motion	None yet — a fixed spark	Buoyant convection (plus a magnetic nudge)
Ends when	The channel goes conductive → arc forms	Arc too long to sustain → extinguishes at top