Is There an Electric Field Within the Cavity of a Polarized Hollow Conductor?

AI Thread Summary
In a polarized hollow metallic conductor, the electric field inside the cavity is zero due to Gauss's Law, which states that if there is no charge enclosed, the net electric flux is zero. The charges on the conductor redistribute to the outer surface, ensuring no electric field exists within the cavity. Even in cases where the conductor has a net charge, as long as there is no charge inside the hollow, the electric field remains absent. The discussion also touches on the potential for tangential electric fields in regions of zero charge density, but these do not disrupt the equilibrium state. Ultimately, the consensus is that the electric field within a hollow conductor at equilibrium is always zero.
tade
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Suppose we have a hollow metallic conductor, just a thin metallic shell forming a large hollow cavity.

It is then polarized by electric charges placed nearby externally.

The equilibrium electric field must be parallel to the surface normals of the shell, there must be no tangential component to the electric field.

However, is it possible for there to be a net electric field within the cavity, and if not, why not?

A second scenario is the hollow conductor not being polarized by external charges, but possessing a net amount of electric charge on itself, and i ask the same question again
 
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Assuming that there is no charge inside the hollow of the conductor then there will be no field in the cavity, by Gauss’ law.
 
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Dale said:
Assuming that there is no charge inside the hollow of the conductor then there will be no field in the cavity, by Gauss’ law.
what about there possibly being a net electric field with zero net enclosed electric flux? or equivalently, a field with zero divergence
 
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tade said:
However, is it possible for there to be a net electric field within the cavity, and if not, why not?
Polarisation of the charges on the sphere will produce a minimum of potential energy (equilibrium situation) where there is no Potential Difference across the sphere (so no internal field). Any difference in potential across the inside of the sphere would result in current flow to cancel it, in the steady state.
If the sphere has finite resistivity and thickness, there can be a time lag for the charge flow to achieve the cancellation and that can detract from the 'screening properties' against RF of a metal box, made of real metal and with resistive joints and seams in it.
 
Tade, why do you ask questions if you're just going to argue with us about the answer?

All tha charge on a conductor moves to the outside. There is no field inside the conductor, and there are no free charges. So there can't be any field in the cavity.
 
Vanadium 50 said:
Tade, why do you ask questions if you're just going to argue with us about the answer?
I'm not sure what you mean, its not like I'm trying to force anyone to accept my opinions, I'm just trying to reason it out. I still have questions about it.
Vanadium 50 said:
All tha charge on a conductor moves to the outside. There is no field inside the conductor, and there are no free charges. So there can't be any field in the cavity.

So, Dale mentioned Gauss' Law; Gauss' Law concerns the enclosed flux or divergence of an electric field pertaining to the local charge density, it might be unable to provide a full description of an electric field.

which is what I was asking Dale about just now.

if all the charge moves to the surface (its a hollow shell anyway), how do we mathematically prove that the field inside the cavity is zero?
 
sophiecentaur said:
Polarisation of the charges on the sphere will produce a minimum of potential energy (equilibrium situation) where there is no Potential Difference across the sphere (so no internal field). Any difference in potential across the inside of the sphere would result in current flow to cancel it, in the steady state.
If the sphere has finite resistivity and thickness, there can be a time lag for the charge flow to achieve the cancellation and that can detract from the 'screening properties' against RF of a metal box, made of real metal and with resistive joints and seams in it.
for clarity, not necessarily a sphere, but any shape

I was wondering the electric field inside a hollow shell conductor might be

Also, I understand why a conductor's surface might be an equipotential surface, with the equilibrium electric field parallel to the surface normals of the shell, with no tangential component to the electric field.

but I was thinking about whether its possible to have equilibrium solutions where there are tangential components in regions where the charge density is zero.

since the charge density is zero, there could be tangential components as they wouldn't upset the equilibrium

additionally, i was thinking about whether certain conductor shapes can make it difficult for there to be tangential electrical forces to cause charge flows, "trapping" the flow of charges and resulting in interesting and unexpected equilibrium solutions
 
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tade said:
additionally, i was thinking about whether certain conductor shapes can make it difficult for there to be tangential electrical forces to cause charge flows, "trapping" the flow of charges and resulting in interesting and unexpected equilibrium solutions
I think you may have too 'pictorial' view of this and there is not an equivalent to a rock pool, sitting above the level of the sea. The conductor would be continuous between your postulated hollow region where charge could be 'trapped' so there is always a path. You could try to commit yourself to actually drawing the layout you have in mind. You would find that it can't be done unless you put a charged object inside your hollow shape and, in that case, the charge distribution outside the object would be the same as if it was in free space. And, in any case, it has nothing to do with the original idea.

PS You are having a problem in reconciling the maths with the actuality. Perhaps a bit of 'faith' that it does work would help you with this - rather than being almost convinced that you have found a hole in the system. i.e. "I must be wrong" and not "I could be right". That could be a bit unsatisfactory for you but it may be a way through. :smile:
 
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sophiecentaur said:
I think you may have too 'pictorial' view of this and there is not an equivalent to a rock pool, sitting above the level of the sea. The conductor would be continuous between your postulated hollow region where charge could be 'trapped' so there is always a path. You could try to commit yourself to actually drawing the layout you have in mind. You would find that it can't be done unless you put a charged object inside your hollow shape and, in that case, the charge distribution outside the object would be the same as if it was in free space. And, in any case, it has nothing to do with the original idea.

PS You are having a problem in reconciling the maths with the actuality. Perhaps a bit of 'faith' that it does work would help you with this - rather than being almost convinced that you have found a hole in the system. i.e. "I must be wrong" and not "I could be right". That could be a bit unsatisfactory for you but it may be a way through. :smile:
I'm not almost convinced of anything, I'm just wondering about aspects of the problem

in that case, there's a mathematical proof of this right?
 
  • #10
tade said:
I'm not almost convinced of anything, I'm just wondering about aspects of the problem

in that case, there's a mathematical proof of this right?
Bound to be. The problem will be in translating what you suggest into a valid mathematical model.
 
  • #11
sophiecentaur said:
Bound to be. The problem will be in translating what you suggest into a valid mathematical model.
i think the statement would be, for whatever shape, prove that the electric field within is always zero and that the surface is always at equipotential, for cases at equilibrium
 
  • #12
How could it not be in equilibrium? If not, then charges would flow.
 
  • #14
sophiecentaur said:
How could it not be in equilibrium? If not, then charges would flow.
yeah, I did mention "for cases at equilibrium"
 
  • #16
tade said:
right, and I asked him a question about Gauss' Law regarding the field within a conductor
which I'm still wondering about
so have you read up on Gauss's Law yet ? and searched through a few other pages yourself ?

we are here to help people, not spoonfeed them, it's nice to see people make an effort :smile:
 
  • #17
davenn said:
so have you read up on Gauss's Law yet ? and searched through a few other pages yourself ?

we are here to help people, not spoonfeed them, it's nice to see people make an effort :smile:
yes, I understand what Gauss' Law entails, this is my question to Dale about the internal electric field:
tade said:
what about there possibly being a net electric field with zero net enclosed electric flux? or equivalently, a field with zero divergence
 
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  • #18
tade said:
what about there possibly being a net electric field with zero net enclosed electric flux? or equivalently, a field with zero divergence

An electric field with zero divergence must be zero itself, by Gauss's Law.
 
  • #19
PeterDonis said:
An electric field with zero divergence must be zero itself, by Gauss's Law.
but I'm thinking, a lone point charge has an electric field some distance away from itself, and that electric field at a distance has zero divergence
 
  • #20
tade said:
i'm thinking, a lone point charge has an electric field some distance away from itself, and that electric field at a distance has zero divergence

You are thinking about it wrong. If you evaluate Gauss's Law in the vacuum region surrounding a point charge over a sphere that does not enclose the charge, then it is zero. In that sense, yes, the vacuum field has "zero divergence".

But if you evaluate Gauss's Law over a sphere that does enclose the charge, you cannot get a zero result. So if you know that Gauss's Law evaluates to zero over any sphere whatever, including a sphere just on the inner surface of the conductor in your OP, you know there cannot be any charge anywhere inside that sphere.
 
  • #21
PeterDonis said:
You are thinking about it wrong. If you evaluate Gauss's Law in the vacuum region surrounding a point charge over a sphere that does not enclose the charge, then it is zero. In that sense, yes, the vacuum field has "zero divergence".

But if you evaluate Gauss's Law over a sphere that does enclose the charge, you cannot get a zero result.

PeterDonis said:
So if you know that Gauss's Law evaluates to zero over any sphere whatever, including a sphere just on the inner surface of the conductor in your OP, you know there cannot be any charge anywhere inside that sphere.
I'm not sure if Gauss' Law does evaluate at zero over any sphere within the hollow of the conductor.

anyway, assuming that there's no charge within the hollow, i would like to know whether there could be an electric field within the hollow
 
  • #22
tade said:
I'm not sure if Gauss' Law does evaluate at zero over any sphere within the hollow of the conductor.

Why not?

tade said:
assuming that there's no charge within the hollow, i would like to know whether there could be an electric field within the hollow

No. Do you understand that Gauss's Law evaluates the electric field over a sphere?
 
  • #23
PeterDonis said:
Why not?
cos that might need to be mathematically proven

PeterDonis said:
No. Do you understand that Gauss's Law evaluates the electric field over a sphere?
I understand that Gauss' Law evaluates the net closed-surface electric flux, so I think it might not be able to provide a full description of an electric field
 
  • #24
tade said:
cos that might need to be mathematically proven

It already is. See below.

tade said:
I understand that Gauss' Law evaluates the net closed-surface electric flux

Yes. And we have already seen that, if there is no charge inside a surface, the net flux through that surface must be zero. So, since we have already assumed that there is no charge inside the hollow of the conductor, Gauss's Law evaluated over any surface inside the hollow must be zero.

tade said:
I think it might not be able to provide a full description of an electric field

The flux through a surface only tells you about the field normal to the surface; it tells you nothing about the field tangent to the surface. Yes, that is true.

But now consider: suppose we pick a surface just at the inner surface of the conductor. We know the flux is zero through that surface, therefore the field normal to the surface is zero. But we also know that the field tangent to that surface is zero, since that must be true at any surface of a conductor. So we know the entire field is zero on that surface, and therefore must be zero inside it as well.
 
  • #25
PeterDonis said:
An electric field with zero divergence must be zero itself, by Gauss's Law.
The Coulomb field
$$\vec{E}=\frac{q}{4 \pi r^3} \vec{r}$$
has zero divergence (and also zero curl btw) for all ##\vec{r} \neq 0## but is nowhere 0 ;-)).
 
  • #26
vanhees71 said:
The Coulomb field

Yes, I covered this case in post #20.
 
  • #27
PeterDonis said:
But now consider: suppose we pick a surface just at the inner surface of the conductor. We know the flux is zero through that surface, therefore the field normal to the surface is zero. But we also know that the field tangent to that surface is zero, since that must be true at any surface of a conductor. So we know the entire field is zero on that surface, and therefore must be zero inside it as well.

from my post #7, "but I was thinking about whether its possible to have equilibrium solutions where there are tangential components in regions where the charge density is zero.

since the charge density is zero, there could be tangential components as they wouldn't upset the equilibrium"

also, for the field normals, is it possible that they are non-zero? but some stick in and others stick out such that the net amount is zero
 
  • #28
If you have a conductor in an electrostatic field, there must not be tangential components of the electric field on its surface (because otherwise there'd be a current along the surface), which implies that the electrostatic potential is constant along this surface. If there's no charge distribution inside the conducting surface a solution of the electrostatic equations inside the conductor is ##\vec{E}=0##, i.e., ##\Phi=\text{const}##, which also fulfills the boundary condition that the surface must be an equipotential surface. Then theorems about potential theory (i.e., the Laplace equation, ##\Delta \Phi=0##) tells you that this is the unique solution of the boundary-value problem of the Laplace equation.

Charges outside the conductor of course shift the electrons along the surface such that it becomes an equipotential surface in the static equilibrium state and the electric-field components normal to the surface are discontinuous there with the jump equal (or in SI units proportional) to the induced surface-charge density.
 
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  • #29
vanhees71 said:
If there's no charge distribution inside the conducting surface a solution of the electrostatic equations inside the conductor is ##\vec{E}=0##, i.e., ##\Phi=\text{const}##

what's this theorem called?
 
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  • #30
I'm not sure, whether it has a specific name.
 
  • #31
vanhees71 said:
I'm not sure, whether it has a specific name.
what terms should i search for to find more info about it?
 
  • #32
See, e.g., A. Sommerfeld, Lectures on theoretical physics, vol. 3, Sect. 5.8.
 
  • #33
vanhees71 said:
See, e.g., A. Sommerfeld, Lectures on theoretical physics, vol. 3, Sect. 5.8.
thanks, I'm reading it, and however, it seems to be about the conservation of energy, which is not involved in my issue
 
  • #34
As you'll see, it is involved! It's a very elegant proof, and it's important physics too!
 
  • #35
vanhees71 said:
As you'll see, it is involved! It's a very elegant proof, and it's important physics too!
damn, I'm too stupid to understand it

could you help adapt it to my problem? :oldbiggrin:
 
  • #36
Have you read the chapter in Sommerfeld's book? What's the specific problem with it?
 
  • #37
vanhees71 said:
Have you read the chapter in Sommerfeld's book? What's the specific problem with it?
the specific problem is that I'm too stupid haha

i'm not sure what exactly it is referring and I'm not sure how to apply it to a thin-shell conductor

i'm also afraid that if i do try to apply it, i'll botch it and do it wrong, so i would like some guiding tips first
 
  • #38
tade said:
since the charge density is zero, there could be tangential components

Not at the surface of a conductor. The tangential electric field there must be zero regardless of whether there is any charge density present.
 
  • #39
tade said:
for the field normals, is it possible that they are non-zero? but some stick in and others stick out such that the net amount is zero

Not over the entire sphere at the inside surface of the conductor, because any field with the property you describe would have to have a component tangent to the sphere at at least one point (actually I think it would have to along at least one closed curve on the sphere), and we know there cannot be any nonzero tangential component anywhere on a sphere at the inside surface of the conductor.
 
  • #40
PeterDonis said:
Not at the surface of a conductor. The tangential electric field there must be zero regardless of whether there is any charge density present.
what's the mathematical principle, because technically such fields won't affect equilibrium
PeterDonis said:
Not over the entire sphere at the inside surface of the conductor, because any field with the property you describe would have to have a component tangent to the sphere at at least one point (actually I think it would have to along at least one closed curve on the sphere), and we know there cannot be any nonzero tangential component anywhere on a sphere at the inside surface of the conductor.
do you know of more documentation on this theorem, thanks
 
  • #41
tade said:
what's the mathematical principle

The definition of a conductor. When you say something is a conductor, you are specifying that the tangential electric field on its surface is zero.

Physically, the reason for this definition is that if there were any tangential electric field at the surface of a conductor, charges would move freely on the surface of the conductor to neutralize the field. So the more precise way to specify this property of a conductor is that in equilibrium there cannot be any tangential electric field at the surface of a conductor. Which for this problem amounts to the same thing.

tade said:
do you know of more documentation on this theorem

It's an application of the mean value theorem. For the surface integral of a vector field over a sphere to be zero, the normal component of the field must be negative (pointing inward) on one part of the sphere and positive (pointing outward) on another part (this is basically what you described). But if the vector field is continuous (which the electric field must be), then there must be some set of points on the sphere where the normal component is zero, in between the negative points and the positive points. I believe, as I said, that that set of points must be a closed curve on the sphere, separating the positive region from the negative region.

Now, at any of the points where the normal component of the field is zero, either (1) the field as a whole must be zero, or (2) the field must have a nonzero tangential component. But if the field as a whole were zero at any point on the sphere (which is the only way to avoid having a nonzero tangential component), it would have to be zero everywhere on the sphere, and thus zero everywhere inside the sphere.
 
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  • #42
PeterDonis said:
Physically, the reason for this definition is that if there were any tangential electric field at the surface of a conductor, charges would move freely on the surface of the conductor to neutralize the field. So the more precise way to specify this property of a conductor is that in equilibrium there cannot be any tangential electric field at the surface of a conductor. Which for this problem amounts to the same thing.
however, tangential fields at regions of zero charge density would not affect equilibrium
PeterDonis said:
It's an application of the mean value theorem. For the surface integral of a vector field over a sphere to be zero, the normal component of the field must be negative (pointing inward) on one part of the sphere and positive (pointing outward) on another part (this is basically what you described). But if the vector field is continuous (which the electric field must be), then there must be some set of points on the sphere where the normal component is zero, in between the negative points and the positive points. I believe, as I said, that that set of points must be a closed curve on the sphere, separating the positive region from the negative region.

Now, at any of the points where the normal component of the field is zero, either (1) the field as a whole must be zero, or (2) the field must have a nonzero tangential component. But if the field as a whole were zero at any point on the sphere (which is the only way to avoid having a nonzero tangential component), it would have to be zero everywhere on the sphere, and thus zero everywhere inside the sphere.

lemme check if i understand it correctly:

if there must be a point where both normal and tangent are zero, all normals everywhere have to be zero as well
 
  • #43
tade said:
tangential fields at regions of zero charge density would not affect equilibrium

You keep stating this as if it's relevant. It's not. We're not talking about general "regions of zero charge density". We're talking specifically about the surface of a conductor.

tade said:
if there must be a point where both normal and tangent are zero, all normals everywhere have to be zero as well

No, if there must be a point on a sphere over which the surface integral is zero, where both normal and tangent are zero, the field as a whole everywhere on and inside the sphere has to be zero.
 
  • #44
PeterDonis said:
You keep stating this as if it's relevant. It's not. We're not talking about general "regions of zero charge density". We're talking specifically about the surface of a conductor.

yeah, its all about conductors. in non-conductors the charges are fixed and there's nothing really interesting to consider.

in conductors, charges can flow, and tangent fields make them do that.

so if there's a region on a conductor with a tangent field but zero charge density, the field has no charges to push around and hence equilibrium is not affected.
PeterDonis said:
No, if there must be a point on a sphere over which the surface integral is zero, where both normal and tangent are zero, the field as a whole everywhere on and inside the sphere has to be zero.
nice, yeah, describing all the necessary conditions

interesting theorem, i would like to see how it was derived, do you know of the info or its name so i can search? thanks :thumbup:
 
  • #45
tade said:
in conductors, charges can flow, and tangent fields make them do that

And in equilibrium, all such charge flow will already have taken place in order to neutralize the tangent fields. So in equilibrium, there must be zero tangent field at the surface of the conductor. And you specified you are looking at equilibrium.

How many times are we going to have to say this before you understand it?
 
  • #46
tade said:
do you know of the info or its name

Unfortunately no, I don't. I have tried a little bit of searching online but haven't found any useful reference.
 
  • #47
PeterDonis said:
And in equilibrium, all such charge flow will already have taken place in order to neutralize the tangent fields. So in equilibrium, there must be zero tangent field at the surface of the conductor. And you specified you are looking at equilibrium.

How many times are we going to have to say this before you understand it?
honestly, i don't think i have ignored any point you have posted, at least, i have not intended to, i have tried to assess all the points thoroughly

I don't think its to neutralize tangent fields specifically, rather, tangent fields where there are charge densities will produce flow, which will continue until there are no tangent fields where there are charge densities, and hence no more flow and no more change

so I don't think it rules out tangent fields at regions of zero charge density
PeterDonis said:
Unfortunately no, I don't. I have tried a little bit of searching online but haven't found any useful reference.
no worries
though know of any short mention of it in a text or something? thanks
 
  • #48
tade said:
tangent fields where there are charge densities will produce flow

And so will tangent fields where there is no net charge density. If there is no net charge density and there is an electric field, the end result will be no electric field and a net charge density, which obviously must be maintained by some means external to the conductor--which is also true of a nonzero tangent field on the surface of the conductor if there is no net charge density.

In other words, the situation you are thinking of--no net charge density but a nonzero electric field tangent to the surface of the conductor--is impossible for a conductor in isolation; it is only possible if the conductor is being subjected to some external field. And the conductor's response to the field will be for charge to flow until the tangent field on its surface is zero and the conductor's surface is polarized, with positively and negatively charged regions. One way of thinking of this is that the separated charges on the surface of the conductor are producing a field that is exactly opposite to the tangent field on the conductor's surface induced by the external field, so the net tangent field on the surface is zero.

More generally, the rule is that in any region where charges can flow freely, a nonzero electric field will produce a flow of charge that continues until the electric field is zero. So inside a conductor, where charges can flow freely, there can be no electric field, period, at equilibrium.

On the surface of a conductor, there can be a normal electric field, because charge cannot flow freely normal to the surface since that would mean charge flowing out of the conductor into the free space surrounding it. But charge can flow freely tangent to the surface, so it will do so until any electric field tangent to the surface has been removed.
 
  • #49
Hi, Tade,

Please check section 5-9 and 5-10 in the link https://www.feynmanlectures.caltech.edu/II_05.html

5-9 The field of a conductor
5-10 The field in a cavity of a conductor

It proved that the charge density in the interior of the conductor must be zero, there must be no electric fields inside the empty cavity, nor any charges on the inside surface as well👍.
 
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