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In an electron gun, the heating filament heats the cathode, releasing electrons by thermionic emission. I've read that "electrons are negatively charged particles and the positively charged cylindrical anode develops a strong electric field that exerts a force on the electrons, accelerating them along the tube". However, I don't think that this explanation is very clear, and I was wondering specifically how the "strong electric field" inside the cylindrical anode is able to accelerate the electrons?

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Just to clarify your question a little bit, are you asking how an electric field accelerates a charged particle? Or are you asking if the cylindrical symmetry of the anode should result in zero electric field? –  Colin K Jun 16 '11 at 18:11
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The acceleration mostly happens in the gap between the cylindrical anode's edge and the cathode. The field becomes uniform inside a relatively long cylinder so there is no acceleration there. I am sure others can elaborate this but I think I found the important part of your question. Some book made an error.

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any electric field will accelerate electrons, according to the Lorentz force law:

$\vec{F} = q \vec{E} + \vec{v} \times \vec{B}$,

where q is the charge of the electron, and $\vec{E}$ is the electric field. I assume there is no magnetic field in this example, so the second term is zero.

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