|Friedrich Paschen electromagnetism
Applying a voltage to electrodes at opposite ends of a gas-filled tube didn’t always make the tube glow. Scientists intending to generate an electric arc across a small gap sometimes got an arc following a long irregular path. Friedrich Paschen tested when the arc begins with different gas pressures and different distances between the electrodes showed that the breakthrough voltage is related to the pressure times the distance showed there’s an optimal distance and pressure for the minimum voltage for each kind of gas.
Townsend discharge doesn’t just happen. Each gas has its own ionization energy, the energy needed to dislodge one of its electrons. An electron must accelerate far enough without hitting anything to reach the ionization energy for a gas. If there isn’t enough space to accelerate or there are too many or two few atoms that it can hit then increasing the voltage might work.
Optimal optimum, minimal minimum
The lowest point of the Paschen curve shows the minimum voltage, which we find by setting the derivative of its equation to zero and solving for X. In real life, the trick is to determine the equation. Paschen discovered Paschen’s law empirically; that is, by experiment, which often requires that you simplify and eliminate noise. In Paschen’s case, we ignore the effect of outside light, and assume that one impact makes one ionization, that atoms recapturing electrons are not factors, and that electron production by the cathode is not affected by temperature or humidity.