Current Density between two infinite plates

[themagiciant95]

Hi, I'm studying the "Child Langmuir law". We have a grounded cathode that is an infinite plane with free electrons, and an anode with a positive voltage V. The text says that the current density J is constant between the two plates for the "Charge conservation principle". I was not able to understand how a constant J implies the charge conservation. Could you help me ?

 

[Chandra Prayaga]

Imagine two infinite parallel planes, numbered 1 and 2, also parallel to the cathode and anode, situated between the two plates. There is some charge density between the two planes, consisting of electrons, which are flowing in a direction perpendicular to the two planes. If the current density J₁ through one plane is different from J₂ through the other, that means that with time, there is a pile up of charge between the two plates.

 

[themagiciant95]

So the current intensity can change ,the important is that J=pv stays constant. Is this correct ?

 

[Chandra Prayaga]

We need to be clear about the use of words. Usually, the word "constant" means that the quantity (in this case current density or charge density) does not change with time. If you want to describe a quantity that is the same at different points in space, you use the word "uniform". In the case that I discussed above, the current density is "uniform", in that it is the same through both parallel planes. If it is not "uniform", but remains "constant" in time, then that will result in a charge, and therefore charge density, which keeps on increasing with time. The charge density is then not constant. If this condition of non-uniform but constant current densities is maintained, then you will violate charge conservation because more and more charge will be piling up between the two planes. To avoid this catastrophe, the current density needs to be uniform.

 

[themagiciant95]

Are you able to show me how obtain that dJ / dz = 0 from the "Charge Conservation Law" by calculations ?
This because, if i apply the Continuity Equation in this case ( we have only the z component), i get :

$$\frac{\partial J_{z}}{\partial z} = -\frac{\partial p}{\partial t}$$

 

[Chandra Prayaga]

I assume the p that you used in your equation refers to the charge density. In a "Steady State," the charge density must be constant (independent of time), so the right hand side is zero, which means that dJ/dz = 0. which in turn means that J is uniform.

 

[themagiciant95]

I assume the p that you used in your equation refers to the charge density. In a "Steady State," the charge density must be constant (independent of time), so the right hand side is zero, which means that dJ/dz = 0. which in turn means that J is uniform.

But why dI/dt = 0 in Steady State ? In fact, if in dt some charges enter in a infinitesimal volume and are immediatly slowed down , in the following dt less charges leaves the infinitesimal volume, so the the charge in the infinitesimal volume increase.
Where am i wrong ?

 

[Chandra Prayaga]

Steady state means things don't change with time.