Forget
electrcity ... Have a torch heated valve. :)
I'd love to see you keep that
flame burning while having a
sufficienltly low gas pressure around the cathode to have a long
enough mean free path for the electrons, Not to mention problems
caused by the ions geenrated by said flame.
Doable.
The glass envelope for the vacuum takes the form of two concentric
cylinders joined at their ends, topologically a torus. The cathode is
another concentric cylinder, this time of metal or whatever, just
barely larger than the inner glass cylinder, with the grids and
anode/plate forming successively larger cylinders. The heater flame
passes through the centre of the cylinder (which is outside the glass
vacuum envelope), heating the cathode by infrared radiation, much as a
conventional heater does. (The glass forming the inner cylinder needs
to be infrared-transparent and probably high-melting-point.)
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