- #1
elgen
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This problem is from Griffiths' book Introduction to Electrodynamics [Problem 2.53 in 4th edition].
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.
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.