Wimshurst Machine Sparks

What is modelled

A Wimshurst machine is an electrostatic influence generator. Each sector passing the brushes carries a small packet of charge to the collectors, which charge the Leyden jars. The jars leak through the air and the insulators, the machine's voltage sags as it charges, and a spark jumps across the gap when the electric field reaches the breakdown strength of air.

Equations

• Short-circuit current Isc = N f qs for N sectors, crank rate f and charge per sector pass q_s.
• Charging C dV/dt = Isc(1 − V/Voc) − V/R, giving V(t) = V∞(1 − e−t/τ) with τ = C / (Isc/Voc + 1/R).
• Breakdown in a uniform field Vb = Eb d with Eb ≈ 3 kV/mm; time between sparks −τ ln(1 − Vb/V∞).
• Energy of one spark ½ C Vb²; the longest possible spark has d = V∞ / Eb.

Checks run in the page

• RK4 integration of the charging equation against the closed-form time.
• Charge and energy bookkeeping while the jars fill.
• The longest spark by bisection on the gap.
• Sector-by-sector pumping against the smooth model.

Honest simplifications

The machine is reduced to a current source with a finite open-circuit voltage; the real self-excitation, brush geometry, corona and humidity effects are not solved. Breakdown uses a uniform-field estimate of 3 kV/mm, whereas real sparks depend on electrode shape, pressure and humidity. The time warp only speeds up the charging, not the drawn disc.

• Press Wide gap, no spark and see the voltage stop short of breakdown, then narrow the gap.
• Press Damp air, tiny gap: the lower leakage resistance sets a lower settling voltage and slower sparks.
• Press Fast crank, then Big jars: more current speeds sparks up, bigger jars slow them and make each spark stronger.