On the derivation of Child-Langmuir law

In summary: So we just take it as a fact.In summary, This problem from Griffiths' book Introduction to Electrodynamics [Problem 2.53 in 4th edition] asks to show the relation between the constant current ##I## and the potential difference ##V_0##, which is ##I\propto V_0^{\frac{3}{2}}## according to the Child-Langmuir law. The conversation discusses using Poisson's equation and the relation between current, charge density, and electron speed to solve for ##V##. It is suggested to use the energy relation and multiply with ##\mathrm{d} V/\mathrm{d} x## for a first integral. The final solution is given as
  • #1
elgen
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This problem is from Griffiths' book Introduction to Electrodynamics [Problem 2.53 in 4th edition].
IMG_4076.jpg


It considers that electrons are emitted from the cathode and move to the anode. This establishes a constant current between the parallel plates. It asks to show that the constant current ##I## and the potential difference ##V_0## have this relation ##I\propto V_0^{\frac{3}{2}}##, the Child-Langmuir law.

I started with Poisson's equation
$$\frac{d^2 V}{dx^2} =- \frac{\rho}{\epsilon_0}$$
The relation among the current ##I##, the charge density ##\rho##, and electron speed ##v## is
$$I = A \rho v$$
where ##A## is the plate area. My understanding is that
$$v=\sqrt{\frac{2x}{m}(-\frac{dV}{dx}q)}$$
where ##m## is the electron mass and ##q## denotes the charge. This leads to the following DE
$$\frac{d^2 V}{dx^2} =- \frac{I}{\epsilon_0 A \sqrt{\frac{2x}{m}(-\frac{dV}{dx}q)}}$$
which I do not know how to proceed to solve for ##V##.

In the meanwhile, I suspect that I need to use ##-\frac{dV}{dx}=V_0/d## on the RHS of Poisson's equation. This leads to
$$V=-\alpha\frac{4}{3}x^{3/2}+\frac{V_0+\alpha\frac{4}{3}d^{3/2}}{d}x$$
where ##\alpha=\frac{I}{\epsilon_0 A \sqrt{2V_0q/md}} ##, and
$$-\frac{dV}{dx}=\alpha 2 x^{1/2}-(V_0+\alpha\frac{4}{3}d^{\frac{3}{2}})/d$$.
Wouldn't this contradict to assumption of ##-\frac{dV}{dx}=V_0/d##? I am confused. Any comments on this derivation is appreciated.
 
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  • #2
elgen said:
My understanding is that
v=2xm(−dVdxq)
where m is the electron mass and q denotes the charge.
[tex]\frac{1}{2}mv^2=qV[/tex]
isn't it though sign of q,V are to be investigated carefully ?
 
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  • #3
Thank you for the pointer.

##q## is the electron charge, so it is negative. To have a positive kinetic energy, ##V## is negative, no? This does not feel right, as I would expect ##V## to remain positive between the plates.

In the meanwhile, when I wrote v=\sqrt{2x/m(-dV/dx)q, I wrongly assumed a constant electron acceleration. From the energy relation, Poisson's equation becomes
$$\frac{d^2V}{dx^2}=-\frac{I}{\epsilon_0 A\sqrt{\frac{2q}{m} V}}$$
which does not yield a solution in terms of elementary functions. What else is missing here?
 
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  • #4
Nothing is missing here except the solution of the differential equation. Hint: Multiply with ##\mathrm{d} V/\mathrm{d} x## for a first integral!
 
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  • #5
Thank you. I got $$V=V_0(x/d)^{\frac{4}{3}}$$ and $$I=\frac{-4\epsilon_0 A}{9d^2}\sqrt{\frac{2q}{m}}V_0^{\frac{3}{2}}$$
 
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  • #6
Is there a way to prove that the current ##I## is independent of time ##t## and position ##x## or we just take it as a fact because that's what the experiment tell us?
 
  • #7
The constant current is given as an experiment result.
 
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FAQ: On the derivation of Child-Langmuir law

1. What is the Child-Langmuir law?

The Child-Langmuir law, also known as the three-halves power law, is a fundamental equation in the field of plasma physics that describes the relationship between the electric field, current density, and space charge in a vacuum.

2. Who derived the Child-Langmuir law?

The Child-Langmuir law was derived independently by American physicist Irving Langmuir and British physicist John Child in the early 1900s. Langmuir's work focused on the application of the law to electron beams, while Child's work focused on its application to ion beams.

3. What is the significance of the Child-Langmuir law?

The Child-Langmuir law is significant because it provides a simple and elegant mathematical description of the behavior of charged particles in a vacuum. It has been applied in various fields, including electron microscopy, plasma physics, and space propulsion.

4. What are the assumptions made in the derivation of the Child-Langmuir law?

The Child-Langmuir law is based on the assumptions that the particles are point charges, the electric field is uniform, and the particles are moving in a straight line. It also assumes that the particles do not interact with each other and that there is no external magnetic field present.

5. Can the Child-Langmuir law be applied to non-vacuum conditions?

No, the Child-Langmuir law is only applicable in vacuum conditions where the particles are not influenced by external factors. In non-vacuum conditions, other factors such as collisions and interactions between particles must be taken into account, making the law invalid.

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