Electric field outside the plates of a capacitor

[zorro]

Is there any electric field outside the region between the plates of a parallel plate capacitor? What about near the edges? Do we neglect that variation in deriving an expression for the field inside?
The positive and negative charges appear only on one side of each plate so I don't think there should be any field outside.

 

[LeonhardEuler]

If there were two infinite parallel planes of opposite charge, there would be a field inside them, but not outside them. You can tell this because, assuming you know the derivation for a single plane of charge, you can find the field for two planes by superposition of the solutions, and the fields of oppositely charged plates cancel outside, but reinforce each other between the plates.

In reality, there is a nonzero field outside the plates of a capacitor because the plates are not infinite. A charged particle near the plates would experience a stronger force from the closer plate that is not totally canceled out by the farther one.

And to answer your other question, edge effects are ignored when deriving the simple expression for the field between parallel plates of charge you are probably referring to, but they can also be taken into account to give a more complicated expression.

 

[zorro]

and the fields of oppositely charged plates cancel outside, but reinforce each other between the plates.

I did not get this point.
Refer the figure:

At the point P (not far away from the positive plate), there is a net electric field towards left.
There is a net electric field towards right at the point Q.
(btw do we consider the charge on one side of negative plate to find the electric field at Q? Or is it isolated from Q?)

but they can also be taken into account to give a more complicated expression.

Wow...we can derive the field near the edges too? Never knew that.

 

[LeonhardEuler]

What I mean is that, at the point P, the field from the positive plate pushes to the left, and the negative one pushed to the right, so the fields tend to cancel in this region. In between the plates, the positive plate pushes to the right and the negative one pulls to the right, so the fields reinforce.

If the plates were infinite, the two plates would completely cancel each other outside the region between the plates.

 

[zorro]

My figure was incorrect. There should not be any positive charges on the left side of the left plate. I understood you now.

In reality, there is a nonzero field outside the plates of a capacitor because the plates are not infinite. A charged particle near the plates would experience a stronger force from the closer plate that is not totally canceled out by the farther one.

Now I don't understand this point
Can't we apply this explanation of yours to the above statement? -

"at the point P, the field from the positive plate pushes to the left, and the negative one pushed to the right, so the fields tend to cancel in this region. In between the plates, the positive plate pushes to the right and the negative one pulls to the right, so the fields reinforce."

 

[LeonhardEuler]

Can't we apply this explanation of yours to the above statement? -

"at the point P, the field from the positive plate pushes to the left, and the negative one pushed to the right, so the fields tend to cancel in this region. In between the plates, the positive plate pushes to the right and the negative one pulls to the right, so the fields reinforce."

Yes, but the key is "tend to cancel". They don't completely cancel for finite plates.

 

[zorro]

The electric field due to a plate of the capacitor is independent of the distance from it (its uniform) provided its not infinite. So if the finite identical plates have uniform charge density, away from the edges outside the capacitor the field should be 0.

Are you saying that non-zero electric field is due to increased/decreased field due to edge effects ?

 

[LeonhardEuler]

The electric field due to a plate of the capacitor is independent of the distance from it (its uniform) provided its not infinite. So if the finite identical plates have uniform charge density, away from the edges outside the capacitor the field should be 0.

Are you saying that non-zero electric field is due to increased/decreased field due to edge effects ?

You have that backwards: The electric field is independent of distance only when the plates are infinite. Otherwise a capacitor on another planet would produce the same effect here as nearby one.

 

[zorro]

By infinite I meant infinite distance from the plates (not infinite plate).
How do you prove that the field at a nearby point outside the capacitor is not 0?

 

[LeonhardEuler]

The case of circular plates is simplest. Suppose the charge is uniformly distributed. Look at the force at some place directly above the center of the plate. There is force contributed by all the imaginary rings around the central point on the plate, and all the force from within one narrow ring is uniformly distributed around that ring. To find the total force we need to integrate along the parameter theta, which is the angle from the normal of the plane, to the edge of the plate. So there will be some function to integrate, which will be the z-component of the force:
F_{tot} = \int_{0}^{\theta_{MAX}}g(\theta)d\theta
We don't need to know what form g has. The only important thing is that each ring contributes something to the force, and the sign is the same. For a positive plate the field will be away from the plate.

Now look at a plate farther down from this one. The integral is exactly the same, but \theta_{MAX} is less. So the force must be less.

 

[zorro]

hmmm...I understood it clearly now.
Thank you very much!