Electric field within the cavity of a hollow conductor polarized by external charges?

[vanhees71]

Have you read the chapter in Sommerfeld's book? What's the specific problem with it?

 

[tade]

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

 

[PeterDonis]

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.

 

[PeterDonis]

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.

 

[tade]

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

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

 

[PeterDonis]

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.

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.

 

[tade]

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

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

 

[PeterDonis]

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.

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.

 

[tade]

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.

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

 

[PeterDonis]

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?

 

[PeterDonis]

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.

 

[tade]

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

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

 

[PeterDonis]

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.

 

[alan123hk]

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.