Electrostatic force between 2 distant parallel plates

[Physicianist]

Homework Statement

The problem is:
If you have 2 rectangular plates 1cmx10cm each, which means that the area is A=10cm²=10^-3m²for each plate; every plate holds a Constant positive charge of Q=+2x10^-6C each; these plates are separated by a distance d=1cm=10^-2m

What is the electrostatic repulsive force between these 2 plates?

Homework Equations

1)I've searched all around the internet and only found the formula for 2 narrow parallel plates (d<<the dimensions of the plates), which is not the case here:
F=Q²/(2Cd)
where C is the capacitance of the capacitor formed between these 2 plates(however these plates do not actually form a capacitor with one another because both have the same charge).
2) C= ε₀xA/d (these plates are separated by air)
3) From 1) and 2) I get this result: F=Q²/(2ε₀xA)
Which means that the electrostatic force between 2 narrow plates doesn't change with a little increase in the distance d between them ??!
4) I've also found somewhere on the internet that when the dimensions of the 2 plates aren't infinite relatively to the distance between them, there is a correction factor you should multiply F with, which (is 1+2d/D) where D is the diameter of the plates if they are circular or the square root of their area if they are rectangular. This factor takes into account something called the edge forces that increase with the distance d between the plates. If I multiply F with this factor, I got F increasing with distance between the plates instead of decreasing (which I think is more logical,isn't it)

That's why I urgenly need someone to tell me if I'm on the right track or not because I'm very confused with all that. Every help is welcomed, especially if someone from the admin or an expert in physics or engineering would like to interfere

The Attempt at a Solution

If I take the formulas above, I get F=227,27N without the presumed correction factor
and F=371N after multiplication of F with the correction factor.

 

[gneill]

Just to add to your misery, can you be sure that the plates retain a uniform charge distribution when they interact?

If you sketch some of the force vectors on little dq force patches on one plate as caused by charge all over the other plate, are there areas where the net force is not directed perpendicular to the plate? Will this induce the charges to move away from a uniform distribution? (Perhaps forming a band around the edge and "ears" at the corners or some such thing).

If the charge distribution were guaranteed to remain uniformly distributed (say if the plates were comprised of a charged insulator), then I might suggest performing a numerical integration to find the net force. Basically Coulomb's Law between dq's over all plate surfaces.

 

[Physicianist]

Thank you gneill for your reply, you made me more aware about the extended complexity of this problem. However, I just want to know if there is an answer to this question. Is the force F much less than the results I obtained or close to them.

In fact, I was asking myself if this question should be put in advanced physics or in introductory physics.

Across all the references I found, all agree that when the distance between plates becomes unneglectable relatively to the dimensions of the plates, the force between them increases.

 

[gabrielbhl]

In my opinio you already have the solution:
the force of these two charged plats are F= q²/2A(epsilon_o) for small distances and just
F=kq²/d² for big distances

 

[Physicianist]

Well, Gabriel. Your answer makes sense but I cannot assimilate a 1cmx10cm plates separated by a distance of only 1cm to 2 distant point charges.
So, you want to tell me that when 2 plates begin to separate from each others, the force F increases at the beginning when the distance is in the same scale as the dimensions of the plates but then rapidly declines when the distance becomes much greater than the dimensions of the plates.

In fact, if I calculate the force F between 2 punctual charges at a distance d and the force F between 2 plates at the same distance d (on the condition that d is too small relatively to the dimensions of the plates), the force between the 2 punctual charges will be always greater.
According to the above, do you, Gabriel, want to tell me that at a distance d (not too but not too big to the dimensions of the plates) the force F will be somewhere in the interval between the force between 2 punctual charges and the force between 2 infinite plates at the same distance d and having the same charge Q?

 

[gabrielbhl]

HEY, the force is not like u had predicted F=Q2/(2ε0xA) , it is only q²/2ε0A, it does not depends of the small distance x...
and yes, the force on this case is q²/2ε0A because the plates behave like an parallel plat capacitor.
As the distance x increases, the force decreases (once the capacitance C decreases) and consequently its energy... until the system become two point charge.