# Why does the theory and practice differ?

Let's consider a vacuum diode with cylindrical electrodes.

A voltage is applied to cathode, so there is an electric field between cathode and anode. Also there is a magnetic field $\mathbf{B}$ directed along electrodes. I have considered the trajectories of electrons transferring from cathode to anode at different values of magnetic field using the fact that the Lorentz force is applied to the electrons: $$\mathbf{F} = e(\mathbf{E} + [\mathbf{v} \times \mathbf{B}]).$$ As can be seen, if we increase the value of $\mathbf{B}$, the trajectory will curl more, but the electrons will get to cathode. At some value of $\mathbf{B}$ the trajectory will be tangent to cathode surface, so electrons will not get to cathode - let's call this value $\mathbf{B}_{critical}$. Subsequent increase of $\mathbf{B}$ will increase the curvature of trajectory.

According to this reasoning, we may consider the relationship between current $I$ through diode and the value of magnetic field $\mathbf{B}$:

$$I= \left\{\begin{matrix} I_0, |\mathbf{B}| < |\mathbf{B}_{critical}| \\ 0, |\mathbf{B}| \geq |\mathbf{B}_{critical}| \end{matrix}\right.$$

But according to the real experiment, the value of $I$ at the neighbourhood of $\mathbf{B}_{critical}$ value does not change suddenly, as shown in the second picture. So, the question is why this is happening?

• What happens if the anode is slightly off center, or the cathode is not perfectly round? Or the magnetic field is not perfectly uniform. Commented May 17, 2018 at 22:48
• And scattering of electrons from electrons and electrons from background gas... And energy spread from thermionic emission... Or, in general, reality always diverges from theory... Commented May 17, 2018 at 22:52
• In theory, theory and practice are the same; in practice, they ain't. Commented May 17, 2018 at 23:08

The short answer is that the theory always assumes some things that are only approximated by real systems. More sophisticated theories makes fewer and less obviously wrong assumptions, but there are still assumptions there.

The comments to the question point out some of the assumptions that have gone into this particular case,

• Cicularity of the parts
• Co-centricitly of the parts
• Uniformity of the magnetic field
• Mono-energetic emission (violated by thermal effects if nothing else)
• Lack of electron-electron scattering
• Perfect vacuum as the stage

and we can quickly add a few more

• Zero electrical fields
• Lack of vibration in the structures
• Steady anode voltage
• And so on...

and the thing to note is that not all of them were introduced to the calculation using the word "assume". That's a hint that you need to be on your toes if you want to identify even a majority of the most important the confounding factors in any given situation.