Maxwell equation at the surface of a conductor - paradox?

[coquelicot]

Assume that we have a conductor of any shape, say a ball of copper. At electrostatic equilibrium, it is well known that the potential inside this conductor is constant, for otherwise free charges would move from points of highest potential to points of lowest potential (this includes the surface of the conductor). This implies in particular that the electric field is null inside the conductor and on its boundary. Now, on the surface of the conductor, Maxwell equations give ${\rm div} \vec E = \rho/\varepsilon$, so $\rho = 0$ everywhere on the surface. These equation are true even if an exterior forcing electrostatic field is applied to the conductor, which means that no charge move when applying an electrostatic field to a conductor; this is in total contradiction to what is teached about Faraday cages (say). So, what is wrong ?

 

[cpsinkule]

ρ is not 0 on the surface. on a perfect conductor ALL of the charge is on the surface. the issue is that the electric field is NOT continuous on the boundary of a perfect conductor, but the potential is. when an external electric field is applied, charges in the surface DO move and the e-field inside a conductor is only 0 for static charge distributions. the charges re-arrange themselves so fast that it is barely noticeable unless the applied external field is not a static field. even then, the field inside a conductor is very very close to 0.

 

[coquelicot]

Thank you for trying to answer. It is clear that you have not answered the question, but only stated commonly accepted facts (I observe nevertheless that your claim according to which the "e-field inside a conductor is only 0 for static charge distributions" contradicts the Faraday cage principle, which is valid in electrodynamic situations as well). In other words, I also know that the charges on the surface of a conductor should move under the influence of an electrostatic field, but as I wrote above, this contradicts Maxwell equation ${\rm div}\vec E = \rho/\varepsilon$ (since the electric field should be 0 on the surface of the conductor). This is the paradox and my question is : what is wrong ?

 

[cpsinkule]

The electric field is not 0 at the surface of the conductor. The charges on the surface of a perfect conductor do not move in an electrostatic field, classically. There is no paradox, you are making the paradox. The discontinuity of E gives you a nonzero Efield at the surface. It's not a difficult integral...

 

[USeptim]

I agree with cpsinkule, I don't see any paradox here.

On the surface of an ideal conductor you will have a charge density which will cause an abrupt change in electric field so inside the charged shell you will have no field although outside there is electric field.

Ideal conductors does not exist since for example charges will have to move infinitely fast in order to instantaneously neutralize changes in the external electric field but ideally they could create null field inside the conductor.

 

[Dale]

@coquelicot you have to stop claiming that a conductor violates Maxwells equations. That is not what PF is about.

Regarding your specific "paradox". Inside the conductor charges are free to move in any direction, so there cannot be an electrostatic E field in any direction. Because the E field is zero everywhere inside the conductor the divergence of the E field is zero and therefore the charge is also zero inside the conductor.

At the surface of the conductor charges are free to move parallel to the surface, so there cannot be an electrostatic E field parallel to the surface.

However, at the surface the charges are not free to move perpendicular to the surface. Therefore there can be an electrostatic E field perpendicular to the surface. If there is such an E field then the divergence of the E field at the surface is not zero and therefore the charge at the surface is not zero.

 

[Confucius00]

I would like to point out that @coquelicot, which is one of my friends, has not claimed that a conductor violates Maxwell law (at least not in this thread), but on the contrary has stated that this is wrong and tried to undertand what seemed to him a paradox (presenting things as an apparent paradox is a common and well accepted way to ask questions) . Actually he is satisfied with the answer of @DaleSpam as it appears that he didn't take into account the orthogonal component of the electric field on the surface of the conductor. So, what was for him a paradox appears now to be a simple mistake, and this was exactely why he asked this question.

 

[PeterDonis]

your claim according to which the "e-field inside a conductor is only 0 for static charge distributions" contradicts the Faraday cage principle, which is valid in electrodynamic situations as well

The "Faraday cage principle" does not say that the e-field inside a conductor must always be zero, even in dynamic situations. If that were true, copper wires could not conduct electric currents, since doing that involves a nonzero (time varying) electric field inside the conducting wire.

 

[Confucius00]

The "Faraday cage principle" does not say that the e-field inside a conductor must always be zero, even in dynamic situations. If that were true, copper wires could not conduct electric currents, since doing that involves a nonzero (time varying) electric field inside the conducting wire.

Your remark is interesting, and illustrates the fact that sufficient litterature about Faraday cages is lacking. I'm not sure that an electrically fed wire can be assimilated to a Faraday cage: it is more a circuit fed with a power supply. Anyway, I think it is well known that a Faraday cage isolates (more or less perfectly, according to the waves frequency and the quality of the cage) its interior from the exterior influence of EM fields, in dynamic situations or not. "Exterior influence" is the topic of this thread, and not circuits with "inner feeding" (I'm not claiming these terms make sense but trying to express what I feel).
On the other hand, if the issue you pointed out is "perfect isolation", I would say equally that the claim according to which "the e-field inside a conductor is 0 in the electrostatic situation" is wrong as well: As coquelicot tried to explain in a previous (erased) thread (without claiming this is a physical truth), this depends upon the free charges that can be contributed by the conductor at any of its parts. Take for example a very very bad conductor : if you apply to it a strong electrostatic field, it is very unlikely (in my opinion) that the field inside the (bad) conductor will cancel.

 

[PeterDonis]

Thread closed for moderation.

Edit: reopened with a reminder to avoid speculation and discussion of moderation topics here.

 

[Confucius00]

@coquelicot
Regarding your specific "paradox". Inside the conductor charges are free to move in any direction, so there cannot be an electrostatic E field in any direction. Because the E field is zero everywhere inside the conductor the divergence of the E field is zero and therefore the charge is also zero inside the conductor.

At the surface of the conductor charges are free to move parallel to the surface, so there cannot be an electrostatic E field parallel to the surface.

However, at the surface the charges are not free to move perpendicular to the surface. Therefore there can be an electrostatic E field perpendicular to the surface. If there is such an E field then the divergence of the E field at the surface is not zero and therefore the charge at the surface is not zero.

@DaleSpam. I was previously satisfied with your answer. Nevertheless, I have now one more question. Let $x,y,z$ be an orthogonal system of axes whose origin is at some point of the surface of the conductor, and such that $x,y$ is parallel to the surface and $z$ is orthogonal to the surface, pointing to the exterior. As you have explained, the electric field is 0 inside the conductor, and its $x,y$ components are nul on the surface of the conductor. So, $\vec E = (0,0,E_z)$ and ${\rm div}\vec E = {\partial E_z\over \partial z}$. Now, either the electric field $\vec E$ is not null at the surface of the conductor, in which case $\vec E$ is not continuous and ${\rm div} \vec E$ makes no sense (at least according to the usual definition of the derivative), or $\vec E=0$ on the surface of the conductor. But in this case too, the left derivative of $E_z$ according to $z$ is equal to 0 (since the e-field is equal to 0 inside the conductor), while the right derivative may or may not be equal to $\rho/\varepsilon$. This means that the orthogonal $z$ component of the e-field is not differentiable at the surface of the conductor, since the left and right derivative are distinct (right ?), and in both cases, apparently, Maxwell equation ${\rm div} \vec E = \rho/\varepsilon$ cannot be asserted, at least without introducing some exterior concept (right ? this is not a personal speculation, I am simply trying to understand). I guess that the solution to this problem may involve distributions, but I can't figure out how. Have you any insight ?

 

[Dale]

Are you familiar with the Dirac delta distribution?

The "function" describing a surface charge is the Dirac delta, whose integral is the discontinuous Heaviside theta function. It has the necessary mathematical properties.

$\nabla \cdot (0,0,H(z))=\delta(z)$

 

[Confucius00]

Are you familiar with the Dirac delta distribution?

The "function" describing a surface charge is the Dirac delta, whose integral is the discontinuous Heaviside theta function. It has the necessary mathematical properties.

$\nabla \cdot (0,0,H(z))=\delta(z)$

Yes, but what does it mean regarding the e-field and Maxwel equations ? does an equation like $\delta(z) = \rho/\varepsilon$ make sense ?

 

[Nugatory]

This means that the orthogonal $z$ component of the e-field is not differentiable at the surface of the conductor, since the left and right derivative are distinct (right ?), and in both cases, apparently, Maxwell equation ${\rm div} \vec E = \rho/\varepsilon$ cannot be asserted...

It can. You're right that we have a problem with $\frac{\partial{E_z}}{\partial{z}}$ at the boundary because $E_z$ is discontinuous there, but that's not fatal because the identification of that partial derivative with the divergence only works where there are no discontinuities. Instead, we can work with $E_z(z-\epsilon)-E_z(z+\epsilon)$ as $\epsilon$ becomes arbitrarily small. This quantity is sensibly defined across the discontinuity.

You will find similar concerns, and will have to deal with them in a similar way, when you apply Maxwell's equations to the electrical field of a point particle in a vacuum. There's no problem taking partial derivatives of the field components away from the charge and getting a zero divergence, but you can't sensibly evaluate the partials at the central point. We can, however, find the non-zero flux through the border of an infinitesimal volume surrounding the central point.

 

[Dale]

Yes, but what does it mean regarding the e-field and Maxwel equations ? does an equation like $\delta(z) = \rho/\varepsilon$ make sense ?

It means that the E field is 0 inside the conductor and normal to the surface outside. It also means that there is charge on the surface, but not inside the conductor. Exactly what we want.

Of course, it is a classical approximation, so it breaks down if you look at such a small scale that quantum effects become important. But at classical scales it is reasonable and valid.

 

[Confucius00]

It can. You're right that we have a problem with $\frac{\partial{E_z}}{\partial{z}}$ at the boundary because $E_z$ is discontinuous there, but that's not fatal because the identification of that partial derivative with the divergence only works where there are no discontinuities. Instead, we can work with $E_z(z-\epsilon)-E_z(z+\epsilon)$ as $\epsilon$ becomes arbitrarily small. This quantity is sensibly defined across the discontinuity..

@Nugatory. I don't see how the symetric derivative can be of any help to give a sense to the divergence in this case. Can you be more precise and show me equations, so I'll be able to grasp it ?

@DaleSpam. More or less, I would say you the same. Can you show me how you introduce the Dirac distribution inside the equations, and deduce from Maxwell laws your assertions ?

 

[Dale]

Can you show me how you introduce the Dirac distribution inside the equations, and deduce from Maxwell laws your assertions ?

It is just an ansatz. Based on experience with the distributions I guess the result and then I check if my guess is correct. That is how a large portion of these problems are solved.

 

[Confucius00]

@DaleSpam. Thank you for your answers. I don't want to make you vasting time, in particular because I'm not sure this problem is of interest for you. You seem to say that you have guessed the form of the solution, and checked that it works. This is this "checking" that would have been interesting for me. But as I've just said, I don't want to make you vasting time and it's ok if you want to give up.

 

[Dale]

@DaleSpamYou seem to say that you have guessed the form of the solution, and checked that it works. This is this "checking" that would have been interesting for me.

Ah, I misunderstood what you were asking.

The checking is relatively straightforward.

$\nabla \cdot (0,0,H(z)) = \frac{\partial}{\partial z} H(z) = \frac{d}{dz} H(z) = \delta(z)$

 

[Confucius00]

Ah, I misunderstood what you were asking.

The checking is relatively straightforward.

$\nabla \cdot (0,0,H(z)) = \frac{\partial}{\partial z} H(z) = \frac{d}{dz} H(z) = \delta(z)$

This is not what I meant. You assume that the e-field is discontinuous, say Heaviside-shaped. Then its derivative along z is Dirac. So far so good. But where are Maxwell equations in all of this ? How do you find the magnitude of the field near the surface, or more precisely, its orthogonal slope in the outer direction ? (if the field were not discontinuous, Maxwell equations would give $\rho/\varepsilon$). Saying that the e-field is Heaviside near the surface without showing that this helps somewhere in equations, amounts to nothing else than an affirmation that can not be confirmed (at least without doing experiments) ; even if this is true, this is more an experimental hypothesis than a theoretical deduction, or than a theoretical hypothesis supported by satisfying theoretical consequences. In particular, this does not prove, apparently, that "there are charges on the surface" as you seem to have asserted above : we reached previously this conclusion from another point of view, using the Lorenz force acting on free charges.

 

[vanhees71]

The point is that classical electrodynamics is a coarse-grained version of the full quantum description of macroscopic phenomena. The latter is very complicated and very often unnecessary. Thus we make some idealizing assumptions, simplifying the description of matter to the macrocopic essence, so to say.

In the process of this idealization, singular charge-density and current-density distributions are used. Let's stick to electrostatics. The Maxwell equations reduce to
$$\vec{\nabla} \times \vec{E}=0, \quad \vec{\nabla} \cdot \vec{D}=\rho, \quad \vec{D}=\epsilon \vec{E},$$
where the medium is described by one scalar function $\epsilon=\epsilon(\vec{x})$, the dielectric constant at each point of the space filled with medium (or vacuum, where no medium is and where $\epsilon=1$ in the here used Heaviside-Lorentz units).

The charge density $\rho$ describes the charges treated as "external", i.e., not bound in the medium, whose response is mapped to the consistuent equation in terms of the dielectric function. This density can be a continuous function, giving the number of charges per volume around each point of space.

Now, if you have surfaces between two media or a medium and the vacuum, nother kind of charge density occurs. You just describe the surface as smooth and then there can surface charges. Using your coordinate system around a point, i.e., $x,y$ axes along the tangent surface of the surface and $z$ perpendicular to the surface, you describe the surface-charge distribution by a function $\sigma=\sigma(x,y,0)$ along the surface. It gives the charge per area in this point. You consider this as sharply located charge on the surface, i.e., a little distance away (no matter whether in or outside the medium) you assume the charge density to be 0. The only way to describe this is to use a Dirac-$\delta$ distribution:
$$\rho(\vec{x})=\sigma(x,y) \delta(z).$$
Note that also the dimensions are correct, because $\sigma$ has the dimension charge/Area and the $\delta$ distribution the dimension 1/length, making the singular $\rho$ describing a surface-charge distribution, of the correct dimension charge/volume.

You can also consider line-distributions. This occurs when you idealize thin objects as simple lines, along which you describe the charge distribution by a density $\lambda$. Let's use local coordinates around a point such that the $x,y$ axes span the plane perpendicular to the wire and the $z$ axis tangent to the wire at this point. Then $\lambda=\lambda(z)$, and the charge density is described by the even "more singular" expression
$$\rho(\vec{x})=\lambda(z) \delta(x) \delta(y).$$
Finally, we have the idealization of a charge distribution located in a very small volume (small compared to the distance where I measure the electric field), which we often idealize as a "point charge". Then you simply have a point with a total charge $Q$ at, say, the origin of a coordinate system. The corresponding charge distribution is a pure $\delta$ distribution,
$$\rho(\vec{x})=Q \delta^{(3)}(\vec{x}).$$
Now, it's important not to forget, that all these singular charge distributions are idealizations, just helping to solve a problem more easily (sometimes even analytically), but that these idealizations have their limitations.

Particularly the idea of a classical point charge is flawed. The Maxwell equations finally break down for this idealization, and you cannot find a fully self-consistent description of the mechanics and electrodynamics of a classical point charge. Only approximations hold, and are quite successful to describe complicated situations, where these approximations are applicable.

Also the above given very simple consituent equation with a simple dielectric function is not the whole truth in all circumstances. E.g., for an anisotropic material like a crystal, $\epsilon$ becomes a tensor $\hat{\epsilon}$ or if you have long-range correlations you have a more complicated constituent equation like
$$\vec{E}(\vec{x})=\int \mathrm{d}^3 \vec{x}' \hat{\epsilon}(\vec{x}-\vec{x}')\vec{D}(\vec{x}').$$
Or for strong external fields, i.e., fields that come close to the strength of the internal fields holding the atoms in the material together, the here applied linear-response theory breaks down, and the relation between $\vec{E}$ and $\vec{D}$ becomes a complicated non-linear function (or functional).

 

[Dale]

This is not what I meant. You assume that the e-field is discontinuous, say Heaviside-shaped. Then its derivative along z is Dirac. So far so good. But where are Maxwell equations in all of this ? How do you find the magnitude of the field near the surface, or more precisely, its orthogonal slope in the outer direction ? (if the field were not discontinuous, Maxwell equations would give $\rho/\varepsilon$). Saying that the e-field is Heaviside near the surface without showing that this helps somewhere in equations, amounts to nothing else than an affirmation that can not be confirmed (at least without doing experiments) ; even if this is true, this is more an experimental hypothesis than a theoretical deduction, or than a theoretical hypothesis supported by satisfying theoretical consequences. In particular, this does not prove, apparently, that "there are charges on the surface" as you seem to have asserted above : we reached previously this conclusion from another point of view, using the Lorenz force acting on free charges.

I am not sure where the disconnect is. Perhaps you are uncomfortable because I have been showing the correct form of the equations and neglecting constants? I usually do that so that the structure of the fields and sources can be clear without all of the unnecessary clutter caused by the choice of units. Let me go through and add in the units so that the connection that I have already shown can be more obvious.

For the electrostatic case $B$ is 0, so we only care about Gauss' law and Faraday's law. In SI units these are:
$\nabla \cdot E = \rho/\epsilon_0$
$\nabla \times E = 0$

From our prior considerations (using the coordinates and geometry you described) we determined that $E=(0,0,E_+ H(z))$, where $E_+$ is the magnitude of the E field just outside of the conductor and H is the Heaviside unit step function. This is our ansatz which we will plug into the above equations to verify that it works.

Evaluating $\nabla \times E = \nabla \times (0,0,E_+ H(z)) = 0$ so this field satisfies Faraday's law.

Evaluating $\nabla \cdot E = \nabla \cdot (0,0,E_+ H(z)) = E_+ \delta(z)$ where $\delta$ is the Dirac delta function. So this field satisfies Gauss' law for a charge distribution of $\rho = \epsilon_0 E_+ \delta(z)$

Is anything still unclear?

Hopefully it is clear that this E field is 0 inside the conductor, and that at the surface of the conductor it is normal to the conductor. This matches our physical reasoning above. Hopefully it is also clear that in order to satisfy Maxwell's equations it leads to the given charge distribution, which is 0 inside and outside the conductor and only non-zero at the surface of the conductor.

 

[Confucius00]

The point is that classical electrodynamics is a coarse-grained version of the full quantum description of macroscopic phenomena. The latter is very complicated and very often unnecessary. Thus we make some idealizing assumptions, simplifying the description of matter to the macrocopic essence, so to say. etc.

@vanhees71. Thank you so many for your explannations, without which the expression $\rho(x,y,z) = \sigma(x,y)\delta(z)$ would have seem so weird to me. You have dealt precisely with the dimensional issue : I think this is a model of answer, about a topic that is hardly found in classical books (I think). Simply one of the best answer I've ever got.

@DaleSpam. Thank you for all, even if your last answer was unnecessary after the explannations of Vanhees71, which got exactely what was confusing me. I hope I have not abused of your patience. This modelization of surface and linear densities with the dirac distribution is an important point, in my opinion, and will probably confuse other persons in the future ; but they can, now, find the answer in this forum.

 

[Dale]

Thank you for all,

You are welcome, and I am glad that vanhees71 was able to pinpoint your confusion.