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

In summary: 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.
  • #1
patric44
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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 ?
 
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  • #2
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.
 
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  • #3
patric44 said:
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##.
 
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  • #4
Delta2 said:
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
 
  • #5
patric44 said:
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.
 
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  • #6
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
 
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  • #7
Lord Jestocost said:
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.
 
  • #8
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##.
 
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  • #9
Lord Jestocost said:
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.
 
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  • #10
Delta2 said:
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.
 
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  • #11
Lord Jestocost said:
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.
 
  • #12
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?
 
  • #13
Lord Jestocost said:
by assuming that a steady state can exist
Not to assume but to prove it somehow (with means other than experimental).
 
  • #14
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.
 
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  • #15
Lord Jestocost said:
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?
 
  • #16
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1. How does the current flow between the plates of a vacuum tube?

The current between the plates of a vacuum tube is created by the flow of electrons from the cathode (negative electrode) to the anode (positive electrode). This flow of electrons is controlled by the electric field created by the voltage difference between the plates.

2. What factors affect the current between the plates of a vacuum tube?

The current between the plates of a vacuum tube is affected by the voltage applied to the plates, the distance between the plates, the material of the plates, and the vacuum level inside the tube. These factors can impact the strength and stability of the electric field and therefore affect the flow of electrons.

3. How does the current between the plates of a vacuum tube change with temperature?

The current between the plates of a vacuum tube is directly proportional to the temperature of the cathode. As the temperature of the cathode increases, the number of electrons emitted also increases, resulting in a higher current between the plates. This is known as thermionic emission.

4. What is the purpose of the current between the plates of a vacuum tube?

The current between the plates of a vacuum tube is used to amplify or control the flow of electrons. This is essential in various electronic devices such as amplifiers, oscillators, and rectifiers. The current can also be used to generate electromagnetic waves for communication purposes.

5. How does the current between the plates of a vacuum tube differ from the current in a solid-state device?

The current between the plates of a vacuum tube is carried by free electrons, while the current in a solid-state device is carried by both free electrons and holes (absence of electrons). Additionally, the current in a vacuum tube is not affected by temperature as much as in solid-state devices, as the latter can be affected by factors such as impurities and crystal structure.

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