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If I use a high voltage DC current in a vacuum tube to bring a lower power AC current into a vacuum tube, how do I calculate the average velocity of the AC electrons?

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Since electrons in a vacuum tube accelerate, their speed is increasing as they get closer to the anode.

The top velocity of the electrons, just before they hit the anode, could be roughly estimated based on the work and energy equation:

$eV = \frac {m\nu^2} 2$

From this equation:

$\nu = \sqrt \frac {2eV} m$

The graph for the top speed is shown below.

enter image description here

If, in addition to a large DC voltage, we apply a small AC voltage, the velocity of the electrons is going to vary around the DC point. Due to a non-linear relationship between the voltage and the velocity, for a given AC voltage, the AC velocity will depend on the $V_{DC}$ level.

For a small AC voltage swing, $\Delta V$, given a DC voltage level $V_{DC}$, we can estimate the peak-to-peak swing of the AC velocity, $\Delta \nu$, as follows:

$\Delta \nu$ = $\bigg(\sqrt \frac {2eV} m\bigg)_{V=V_{DC}}'\Delta V = 0.5\sqrt \frac m {2eV_{DC}} \Delta V$

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