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A planar diode in a vacuum bulb can support current in only one direction. To even get current flowing, you have to heat up the cathode filament to excite the electrons. However, once current is flowing, there is a certain limit imposed on it, making the resulting graph of current versus voltage look like this:

Graph of Current vs Vooltage

You can observe an inflection point about half way through the graph, and you'll notice the curve slowly flattening out.

What is the formal theory and terminology behind this limit? Is it because of the build up of space charge (slow moving electrons, in this case) in between the diodes, which would repel the electrons in the cathode and keep them from escaping across the diode, thus imposing a certain limit on current?

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I think the inflection is the transition from Childs-Langmuir space-charge-limited emission (I=PV^3/2) at the lower currents to temperature-limited thermionic emission.

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