Electric field outside 2 same charge parallel plates

[Taniaz]

Homework Statement

Given 2 positively charged parallel plates with equal surface charge densities, calculate the electric field at a point outside the the parallel plates.

Homework Equations

E= (surface charge density) / 2(relative permittivity) for a single sheet.

The Attempt at a Solution

What I'm confused about is if I have 2 positively charged parallel plates with the same surface charge density, and there exists no field in between because the fields point in the opposite direction so they vectorially cancel out.
At a point outside the plates, why is there a field due to both plates? Won't there only be a field at that point due to the closest plate? The field due to the further plate won't reach this point since it diverges upwards due to repulsion?

 

[haruspex]

The field due to the further plate won't reach this point since it diverges upwards due to repulsion?

The field lines for the total field will diverge, but you can always find the total field merely by summing the fields from the individual charges, these behaving as though no other charges are present.

 

[andrewkirk]

The field between the two plates will only be zero at a single point between the two plates. That point is equidistant between the two plates, in the line between the centres of charge of the two plates. At a point closer to one plate than the other the field will have a component pointing towards the further plate and a component pointing away from the zero point. At a point equidistant between the two plates and away from the centre of charge axis, the field will point directly away from that axis, parallel to the two plates.

At a point outside the two plates, just as inside, the electric field is the vector sum of the fields attributable to each of the two plates. That field will point in a direction that is approximately heading away from the zero point described above. If you draw a diagram with the flow lines, you will see that - except where it touches a capacitor plate - the field between the plates phases continuously into the field outside the plates. It must do that because the two electric fields that sum to give the combined field are continuous functions of location everywhere except where they meet a plate.

 

[conscience]

The field between the two plates will only be zero at a single point between the two plates. That point is equidistant between the two plates, in the line between the centres of charge of the two plates. At a point closer to one plate than the other the field will have a component pointing towards the further plate and a component pointing away from the zero point.

This is incorrect .

Net electric field is zero in the entire space between two large uniformly charged parallel plates having equal positive charges .

 

[haruspex]

This is incorrect .

No, what Andrew posted was completely accurate. That said, if the plates are very wide compared with the distance between them, and compared to the displacement of the point from the central axis, then the field at such a point between them is negligible.

In the present thread, it is not stated that the plates are large, nor that the point of interest is close to the central axis, but it probably should have.

 

[conscience]

In the present thread, it is not stated that the plates are large

Homework Equations

E= (surface charge density) / 2(relative permittivity) for a single sheet.

 

[haruspex]

Homework Equations

E= (surface charge density) / 2(relative permittivity) for a single sheet.

In the present thread, it is not stated that the plates are large

Ok, I'll clarify that: In the problem as presented in the present thread, it is not stated that the plates are large.
The relevant equation is provided by the OP, who is therefore making the reasonable assumption that the problem should have stated the plates are large, etc. The fact remains, the problem statement, as presented, does not specify that.

More to the point, Andrew's explanation was correct. Please acknowledge that.

 

[conscience]

More to the point, Andrew's explanation was correct. Please acknowledge that.

Yes . The explanation is correct if plates are not large

Generally in Intro Physics problems plates of parallel plates are implicitly assumed to be large . Even the relevant equations by the OP supported this assumption .

 

[haruspex]

The explanation is correct if plates are not large

It is correct for any size of plate. It is merely that the zero field approximation is valid when the plates are large relative to both the separation of the plates and the distance of the point from the central axis.

 

[andrewkirk]

Yes . The explanation is correct if plates are not large

According to my reasoning, the 'if the plates are large' qualification is unnecessary, for two reasons:

First consider a positive charge on the central axis and very close to plate A. Its repulsion from A will be stronger than its repulsion from B, so it will be accelerated towards B. So along that central axis there is a nonzero field that everywhere points towards the midpoint. This applies regardless of the size of the plates. If intro physics texts say the field is zero, I assume they are referring to the average field along a line between the two plates.

Second, consider a test positive charge between and equidistant from the two plates, that is only a tiny distance inside the outer boundary of the two plates. This charge will be pushed towards the outside of the capacitor, along the plane that is midway between the plates. So there is an electric field at that point, directed towards the outside of the capacitor. As we go further into the capacitor, that field reduces, finally reaching zero at the midpoint of the central axis. If the plates are large, the field will be small in most places on the plane midway between the plates, because most of that is well away from the boundary. So again, if intro physics books say the field is zero, they are probably referring to an average over the full area of the capacitor.

In fact, it will be exactly true (no approximation needed) that the integral of the electrical field over the volume enclosed by the two plates is zero. But the local field will not be zero, or negligible, at points away from the central plane, or near the boundary of the capacitor, and that applies regardless of how large the plates are.

Since the OP was asking about the field at a point outside the plates, I inferred that he was puzzling over an apparent lack of continuity of the field near the boundary of the enclosed region (which puzzled me too when I read the question). That lack of continuity is implied by accepting the 'zero field between the plates' assumption, and can only be dismissed by discarding that assumption and recognising that the field near the capacitor boundary is nontrivial, regardless of the plate size.

 

[haruspex]

Since the OP was asking about the field at a point outside the plates, I inferred that he was puzzling over an apparent lack of continuity of the field near the boundary of the enclosed region

I think it is clear that the question concerns a point outside the plates in the sense that one of the plates lies between that point and the other plate. Indeed, it could have avoided the issue of "large plate approximation" by specifying infinite plates.
As I read the OP's question, Taniaz understands this. Rather, Taniaz has a misunderstanding about how fields interact, thinking that the nearer plate will not only generate a field at the point in question but also mask the field from the further plate.