Why is there a need to average the electric field between capacitor plates?

[center o bass]

As I have understood it, it is a classical problem to show that the force between to capacitorplates is $$\frac{1}{2}\epsilon _0 E^2 A$$, which is at first counterintuitive since one should think that it is $$QE = \epsilon_0 E^2 A$$. I am pondering that question a bit now and i saw the MIT lecture http://ocw.mit.edu/courses/physics/8-02-electricity-and-magnetism-spring-2002/video-lectures/embed07/ where it is explained, but I don't quite get the explanation. When is it ever assumed that the conductor has any with? That is.. is it not only a plane in which it does not have a volume and therefore it is absurd to talk about the electric field inside the conductor?

And why would we ever have to average the two anayway? The efield inside doesn't apply any force. The only force on the plate is from the efield outside and that is certainly E.

Can anyone explain this to me?

 

[aim1732]

You would most certainly agree that the electric field outside the plate is due to both plates. You logic is wrong because a source (even extended one like a plate) can not apply force on itself due to its own field.One simple way to look at this is to assume that the plates have an individual field of E1 an E2. Then inside the plate E1-E2 =0 and outside E1+E2=σ/ε. Solve them and use the field of the other plate to calculate the force on the plate in question.
This result,valid for an infinite plate,is also used to calculate force on the points on a charged conductor due to obvious reasons.

 

[center o bass]

Ah! So what you're really saying is that it's all solved with the fact that the plates can only apply forces to each other?

I have indeed assumed that the total field inside the capacitor exerts a force on a plate, but half of this field is ofcourse due to one plate itself... so ofcourse a particle moving inside that field would experience a force equal to qE, but the plates them selves does ofcourse only experience 0.5QE from each other. Right?

 

[aim1732]

Yes right.

 

[paranormal]

I can provide an explanation for the force on capacitor plates. First, it is important to understand that the force between capacitor plates is due to the electric field created by the charges on the plates. This electric field exerts a force on the charges, causing them to move towards or away from each other, depending on their polarity.

Now, let's break down the equation \frac{1}{2}\epsilon _0 E^2 A. This equation represents the force per unit area between the plates, with \epsilon_0 being the permittivity of free space, E being the electric field strength, and A being the area of the plates. This is known as the electric pressure, and it is the force that the electric field exerts on the charges.

The confusion may arise from the fact that the electric field inside a conductor is zero. This is because conductors have free electrons that can move freely in response to an external electric field, canceling out the field inside the conductor. However, this does not mean that the electric field outside the conductor is zero. In fact, the electric field outside the conductor is what creates the force on the plates.

As for why we need to average the two plates, it is because the electric field is not constant between the plates. The electric field is stronger near the edges of the plates and weaker near the center. In order to accurately calculate the force between the plates, we need to take into account the varying strength of the electric field. By averaging the electric field between the plates, we can get a more accurate representation of the force exerted on the plates.

In summary, the force on capacitor plates is due to the electric field created by the charges on the plates. The electric field outside the conductor is what exerts the force, and the averaging of the electric field accounts for the varying strength between the plates. I hope this explanation helps clarify the concept for you.