Electron tube is an early device that was and still used to study the motion of electron which is a charged particle. Electron tube is a tube with two electrodes(cathode and anode) fitted at the two ends and the tube itself is either vacuum or filled with some gas. If the electron tube is vacuumed then it is called electron vacuum tube and if it filled with gas then it is called gas tube. A figure of electron tube is shown below.

Electrical voltage is between cathode and anode. The cathode is electrical negative and is the source of electrons. The anode is positive with respect to the cathode. Due to the voltage difference between these electrodes, electric field is created within the tube. The electrons fells the electric field and are are forced to move towards to the anode.

To control the amount of electron flowing from cathode to anode, control grids are often used within the tube. If a single control grid is used and the tube is a vaccum the whole device is called triode. A picture of triode is shown below.


The dimension of the tube is very large compared to the electrons or charged particles in the tube. Thus in limits the charged particles electrons can be considered point particles. If q is the charge of an electron then the force F experienced by one charged particles due to the presence of electric field intensity E in the tube is give by,

F = qE

If V is the potential at different points with the tube then we can rewrite above equation as,

F = - q grad.V

From newton law of motion we also have,

F = ma

where m is the electron mass and a is the acceleration.

Let L be the displacement of the particle from arbitrary origin then we can rewrite the newton equation as,

F = m d2L/dt2

Because the tube is evacuated the only force experienced by the electron is due to the electrostatic field present inside the tube. We ignore the magnetic field or gravitational field that might be present.

 We can now equate the two equation for the force and write,

F = qE = m d2L/dt2

 adjusting,

d2L/dt2 = q/m E

If Ex, Ey and Ez are components of E in the x, y and z cartesian coordinate system then we can write,

d2x/dt2 = q/m Ex,
d2y/dt2 = q/m Ey,
and,
d2z/dt2 = q/m Ez

 Again using the fact that E is gradient of potential V we can rewrite,

d2z/dt2 = q/m δVx/δx
d2z/dt2 = q/m δVy/δy
and,
d2z/dt2 = q/m δVz/δz

where δ means partial derivative

If we want to determine the motion of the electrons then we must have knowledge about the initial position, velocity and the potential distribution in the tube. The difficulty here is the field might not be uniform and the at least one of the field component varies with coordinates. This causes the equations to be non-linear and the equations are difficult to solve or the result will be just incorrect. Another difficulty is comes from the speed of the electrons. If the speed of the electrons is much less than speed of light then mass will have to be taken into account and the equations are again non-linear. One way to circumvent the problem with non-linear equation is to use graphical, numerical methods.

There is better alternative method to determine the motion of electrons.

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