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An electrical discharge transfers electric charge through a medium or gap. A gas discharge can begin when accelerated charge carriers produce enough further ionization to sustain an electric current. An electron avalanche describes multiplication by successive ionizing collisions; the supply and loss processes determine whether it becomes a sustained conducting state.

A motivating gas-breakdown experiment varies pressure while keeping the gas, electrode separation and electrode geometry specified. The required electric potential difference does not generally decrease monotonically with pressure. At very low density, an electron encounters too few molecules to sustain multiplication; at higher density, collisions can interrupt acceleration before sufficient ionizing energy is acquired. The resulting dependence on pressure and gap is summarized, within its assumptions, by the Paschen law.^[ref]

The experimental apparatus therefore needs electrodes with a known separation, pressure control and measurement, and a current-limited electrical source with current and potential measurements. Geometry and electrode surfaces must be recorded along with the gas conditions. A measured onset threshold and the later current–potential characteristic answer different questions; the existence of a glowing plasma is not itself a measurement of its onset threshold.

A gas discharge is electrical conduction through a gas containing mobile charged particles. An applied electric field accelerates electrons; collisions can cause ionization, generating additional electrons and ions. A sustained discharge requires charge-production and electrode-emission processes to offset losses, rather than merely the presence of a few initially charged particles.

In a uniform Townsend avalanche model, $d{n}_e/dx=\alpha n_e$ gives $n_e(d)=n_e(0)e^{\alpha d}$ across a gap $d$. If each arriving positive ion releases on average $\gamma$ electrons from the cathode, the self-sustaining threshold is $\gamma (e^{\alpha d}-1)=1$. Combining that condition with the field dependence of $\alpha$ gives the Paschen law.^[ref]

A motivating measurement fixes the gas, electrode material and gap, varies pressure and records the onset electric potential difference with a current-limited source. A pressure gauge and current measurement distinguish the breakdown threshold from the later conducting regime. At larger currents, space charge and heating can invalidate the uniform-field avalanche approximation.

The Paschen law models gas-breakdown electric potential difference as a function of pressure times electrode spacing. For a uniform gap $d$, fixed gas conditions and effective secondary-emission coefficient $\gamma$, an empirical ionization law $\alpha =Ap\exp (-Bp/E)$ together with $\gamma (e^{\alpha d}-1)=1$ givesHere $E=V_B/d$, $A$ has units inverse to pressure times length, and $Bpd$ has units of potential. The denominator must be positive. These constants and the fitted range are part of the model.^[ref]

Writing $L=\log (1+1/\gamma )$ and $x=pd$ gives $V_B=Bx/\log (Ax/L)$. Differentiation places its minimum at $x=eL/A$, with $V_{B,\min }=BeL/A$. A pressure sweep at fixed gap can therefore first lower and then raise the threshold. Nonuniform fields, changing cathode surfaces, temperature changes or microscopic gaps can break this scaling; it is not a universal law for every electrical spark.