Questions related to the current between the plates of a vaccum tube?

[patric44]

Homework Statement

find the current between the plates of a vacuum tube ?

Relevant Equations

J = ρv = -J_{CL}

hi guys
i have an assigment of deriving the current between the plates of a vacuum tube as a function of the potential bias applied on the upper plate
from what i had found this relation is called Child-Langmuir law and states that :
$$I = K V_{d}^{3/2}$$
from what i find online i can derive it from poisson equation $\nabla^{2}\phi = \frac{-\rho}{\epsilon_{o}}$ but i don't get the intuition behind it , i kinda have a sloppy argument not sure if it is true or not : we try to find the potential in the regions between the plates related to current density $\rho$ that comes directly from the heated filament . but then some question in head asks what about the charge density in the two plates ?! did it come directly from the current density $j$ that was released from the filament or it has nothing to do with my solution .
my understanding is a little bit shaky i want some one to explain it a little further .

1) another question :
a paper i read online about Child-Langmuir law states at the top of the page that $J = \rho v = -J_{CL}$ what is that mean! and shouldn't the child-Langmuir current moves that same way as the electrons from the negative plate to the positive one ?!

2) another question that i have no answer to : how the electrons from the filament moves in the first place overcoming the negative repulsion of the cathode in the vacuum tube ?

 

[Delta2]

Are you talking about the derivation of the law from Poisson's equation at this site
https://simion.com/definition/childs_law.html
It seems correct , as regarding all the steps solving a differential equation with boundary conditions, but I also don't get the intuition behind it because there is no figure supplied and also because of my lack of knowledge on the subject of vacuum tubes.

 

[Delta2]

Homework Statement:: find the current between the plates of a vacuum tube ?
Relevant Equations:: J = ρv = -J_{CL}

but then some question in head asks what about the charge density in the two plates ?! did it come directly from the current density j that was released from the filament or it has nothing to do with my solution

As i said i don't know much about vacuum tubes but i think that the answer to the above question might be:
No , the charge density in the two plates doesn't relate to the aforementioned current density j, it is imposed in the problem by the boundary conditions $V(0)=0, V(d)=V_d$.

 

[patric44]

As i said i don't know much about vacuum tubes but i think that the answer to the above question might be:
No , the charge density in the two plates doesn't relate to the aforementioned current density j, it is imposed in the problem by the boundary conditions $V(0)=0, V(d)=V_d$.

so we might say that we are just looking for the solution of the potential between the plates subject to these boundary conditions , unfortunately i can't find other resources regarding Child-Langmuir law online

 

[Delta2]

so we might say that we are just looking for the solution of the potential between the plates subject to these boundary conditions , unfortunately i can't find other resources regarding Child-Langmuir law online

Yes. If it was only the charge density of the plates (equivalently if we neglect the charge density due to the stream of electrons in the vacuum between the plates) then the potential between the plates would be $V(x)=\frac{V_d}{d}x$, and we would find a different relationship for the current between the plates that would depend on x (i think). But it seems to me that according to vacuum tube theory this current is independent of x (we take it to be part of the constant A when we solve the differential equation, hence we consider it to be independent of x), and hence it is just not right to neglect the charge density due to the electrons in the vacuum.

 

[Lord Jestocost]

As the electrons move between the plates with finite velocity, they constitute a space charge region. For the derivation of the Child-Langmuir law, I recommend to have a look at section 4.7.1 The Vacuum Diode in:

Poisson's Equation
faculty.washington.edu › reading

 

[Delta2]

As the electrons move between the plates with finite velocity, they constitute a space charge region. For the derivation of the Child-Langmuir law, I recommend to have a look at section 4.7.1 The Vacuum Diode in:
Poisson's Equation
faculty.washington.edu › reading

That's a very nice pdf and thanks again, but still it doesn't say why we consider the current density J in between the plates to be independent of x.

 

[Lord Jestocost]

Here we are dealing with a non-equilibrium system in a steady state. In case the current density would depend on the spatial coordinate, $x$, this would result in a change in the space charge density distribution between the plates, locally increasing or decreasing (continuity equation). In turn, this would lead to a change in the electric field distribution between the plates with time. The system would thus run in an unsteady state until the space charge distribution (and the electric field distribution) has adapted in such a way that the current density becomes independent of the spatial coordinate, $x$.

 

[Delta2]

Here we are dealing with a non-equilibrium system in a steady state.

How do we know that the system reaches a steady state?

In case the current density would depend on the spatial coordinate, $x$, this would result in a change in the space charge density distribution between the plates, locally increasing or decreasing (continuity equation). In turn, this would lead to a change in the electric field distribution between the plates with time. The system would thus run in an unsteady state until the space charge distribution (and the electric field distribution) has adapted in such a way that the current density becomes independent of the spatial coordinate, $x$.

This argument sounds correct, i just have my doubts regarding the system be in a steady state.

 

[Lord Jestocost]

How do we know that the system reaches a steady state?

Experimentally, when - after applying a given voltage to the vacuum diode - the magnitude of the current delivered by the voltage source no longer changes with time.

 

[Delta2]

Experimentally, when - after applying a given voltage to the vacuum diode - the magnitude of the current delivered by the voltage source no longer changes with time.

Well ok i thought that we might get this by experiment, but i was looking for some sort of theoretical answer.

 

[Lord Jestocost]

A theoretical answer with respect to what? You derive the Child-Langmuir law by assuming that a steady state can exist and you find a self-consistent answer by considering all relevant physical relations. What else are you looking for?

 

[Delta2]

by assuming that a steady state can exist

Not to assume but to prove it somehow (with means other than experimental).

 

[Lord Jestocost]

In case that counts as a proof (somehow) for you, you can solve for the time-dependent current-voltage characteristics of a Child-Langmuir diode by means of analytical or numerical methods.

 

[Delta2]

In case that counts as a proof (somehow) for you, you can solve for the time-dependent current-voltage characteristics of a Child-Langmuir diode by means of analytical or numerical methods.

This seems like a cumbersome task, i guess one would have to utilize the full four Maxwell's equations and not only Gauss's law (which gives us the Poisson's equation). Do you have any books/papers to suggest for this?

 

[Gordianus]

Vacuum Tube Electronics at Ultra-High Frequencies
DOI: 10.1109/JRPROC.1933.227513