Mathematical derivation of the Faraday cage from the Maxwell Equations

[mma]

Hi,

We know that in a space region free from electric charges and surrounded by a conducting surface, the electric field must be zero (this is the Faraday cage).

I suspect that this statement can be derived directly from the Maxwell equations, but I don't find this derivation anywhere. Could somebody show me that?

 

[mma]

Is this question so simple, that it had to be moved to Introductory Physics? :(

Then why does nobody answer me?

 

[mma]

I'm afraid that this isn't trivial. I've found only this guidance about it http://www.pa.msu.edu/~duxbury/courses/phy294H/lectures/lecture6/lecture6.html" :

if there are no charges inside a closed metal surface, then the electric field inside the metal surface is zero everywhere. This is a much more general result than the shell theorems which apply only to uniform spherical shells of charge. Proving this result is beyond the level of this course, so I have to use the dreaded phrase ``it can be shown''.

So "it can be shown". But, how?

 

[Astronuc]

Well trivially, it's Gauss's law. Inside the hollow conductor there is not charge, so the enclosed charge is zero, so the electric field is zero everywhere.

Now more directly, consider the most trivial case of the center of a hollow sphere, with 'uniform' charge on the surface. For each charge, there is an equal charge diametrically opposed, and therefore at the center the electrical fields (vectors) are equal and opposite, so they cancel.

Now, consider any point, off-center. One cannot apply the opposite point charge, but rather one must consider opposing surfaces, dA, which would have charges $\sigma_1\,dA_1$ and $\sigma_2\,dA_2$. Now think if two cones with vertices touching (and having same solid angles) and colinear (parallel) axes, with heights $r_1$ and $r_2$. The E from one is just $\sigma_1\,dA_1$/$r_1^2$ and the other is $\sigma_2\,dA_2$/$r_2^2$, but realize that $dA_i$ is proportional to $r_i^2\,d\Omega_i$, where $d\Omega$ is the solid angle enveloped by cones and subtended by $dA_i$.

So E_i is proportional to 1/$r_i^2$, and $dA_i$ is proportional to $r_i^2$, and the term cancel which then leaves equal charges ($\sigma\,d\Omega$) opposing each other, and therefore the electric fields cancel, i.e. $\vec{E}\,=\,0$.

Voila!

 

[mma]

Well trivially, it's Gauss's law. Inside the hollow conductor there is not charge, so the enclosed charge is zero, so the electric field is zero everywhere.

Not too good. This "proof" doesn't use the presence of the conductive surface. So, it states that the electric field in every chargeless volume vanishes, what is obviously false.


The other solution is applicable only in this very special case :(

But what is the general solution?

 

[Astronuc]

The other solution is applicable only in this very special case.

No, the other solution that I provided ends up being a surface integral. If one makes $d\Omega$ very small and integrates the E field over all $\Omega$, one discovers that the entire integral vanishes, for any arbitrary surface enclosing a volume. All the example shows is that there is an equal charge on one side of the volume opposing the other charge, and the net E field is zero.

For non-spherical surfaces, and particularly nonsymmetric surfaces/volumes, the integrals can be more complicated.

 

[mma]

OK, your solution is good for any shape.
But only for

• uniform charge distribution on the surface
• zero external electric field

The point is that Faraday cage precludes external electric field to enter into its interior.
Evidently, the exterior electric field affects the distribution of the free charges on the metal surface such way that the algebraic sum of the electric field caused by these free charges and the external electric field is zero. I am curious of the proof of this statement.

 

[mma]

For we know that the electrostatic potential on a closed conducting surface is always constant, and the electric field is the gradient of the potential, my question sounds mathematically:

Let's proof that if the potential on a closed surface is constant, and if there is no singularity of the potential inside the region bounded by this surface, then the potential in this whole inner region is constant too.

 

[mma]

Let's proof that if the potential on a closed surface is constant, and if there is no singularity of the potential inside the region bounded by this surface, then the potential in this whole inner region is constant too.

I'm afraid that this isn't true in general. A simple counterexample the V(r)=r potential, and a sphere as the conducting surface.

So, we need some extra conditions about this potential, of course on the bases on the Maxwell's equations.

Any idea?

 

[clive]

If the Astronuc's demonstration didn't satisfy you, then try to solve the Laplacian equation
$$\bigtriangledown^2 V=0$$
with
$$V=V_0$$
on the boundary (using Femlab for example). This Eq. is obviously derived from Maxwell Eqs...

The "counterexample" $$V(r)=r$$ you proposed is wrong because it does not satisfy the equations above.

 

[mma]

If the Astronuc's demonstration didn't satisfy you, then try to solve the Laplacian equation
$$\bigtriangledown^2 V=0$$
with
$$V=V_0$$
on the boundary (using Femlab for example). This Eq. is obviously derived from Maxwell Eqs...

The "counterexample" $$V(r)=r$$ you proposed is wrong because it does not satisfy the equations above.

OK, this seems better. Unfortunately, I am not an expert in PDE-s, and I haven't find the solution of the Laplace's equations for this special case (3-dimensions and constant as boundary condition). I am not curious of of simulations, but only of exact mathematical solution. Can anybody show me the solution of the Dirichlet-problem of the Laplace's equation in 3 dimensions with constant boundary condition?
(of course I see that constant is a solution, but I don't know, why must it be unique)

 

[clive]

You can "feel" what's happen with the following 1-D example:
$$\frac{d^2 V}{dx^2}=0$$ in $$[a,b]$$
and
$$V(a)=V(b)=V_0$$.

(the potential between two infinite metallic plates).
From the first Eq. you have
$$V(x)=mx+n$$
and after imposing the boundary conditions
$$ma+n=V_0$$
$$mb+n=V_0$$
you get $$m=0$$ and $$n=V_0$$.
So the only solution you get is $$V(x)=V_0$$.

Hope it helps...

 

[mma]

Yes, thanks. This is exactly what I thought after your first reply.

 

[mma]

However, thinking in 1 or 2 dimensions instead of 3 can be very misleading.
3-dimensional Euclidean space has completely different properties as 1 or 2-dimensional ones. E.g. the Banach-Tarski paradox is valid in 3 dimension, but not in 2 or 1 (this paradox states that you can cut a solid sphere into finitely many pieces and moving this parts, you can build 2 solid sphere with same size than the original one). Or, in physics, Zeeman's "Causality Implies the Lorentz Group" theorem is valid only in Minkoswski spaces having dimension more than 2.

So we really need a valid uniqueness theorem for the solution of our Dirichlet problem.

 

[mma]

So we really need a valid uniqueness theorem for the solution of our Dirichlet problem.

I've found the proof of the uniqueness. This is the following.

Suppose we have two different solutions of our Dirichlet-problem.
For the Laplace equation is linear and the values of these functions on the boundary are equal, their difference is a solution of the Laplace equation with boundary condition 0. This implies that this difference function is 0, otherwise it would have a local minimum or maximum in the inner region, what contradicts to the fact that it is a solution of the Laplace equation. So the difference of our two solutions is zero, hence these two solutions are equal.
q.e.d.

 

[mma]

But we are not ready yet.

Laplacian equation is valid only in electrostatics.

How can we proof that Faraday cage shields electromagnetic waves as well?

 

[lightgrav]

In the physical world, charge is a property that is intimately connected to ponderous objects. The sluggish response of the charge carriers in a conductor make it possible for microwaves to resonate in an empty cylinder.

 

[mma]

In the physical world, charge is a property that is intimately connected to ponderous objects. The sluggish response of the charge carriers in a conductor make it possible for microwaves to resonate in an empty cylinder.

Does this mean thet Faraday cage shields only static fields? How can then it prevent us e. g. from lightning?

 

[Astronuc]

Does this mean thet Faraday cage shields only static fields? How can then it prevent us e. g. from lightning?

The current travels in the conductive material, not in the hollow (which is not conducting) and travels primarily in the outside surface of the conductor.

 

[mma]

The current travels in the conductive material, not in the hollow (which is not conducting) and travels primarily in the outside surface of the conductor.

OK, but how follows this from the Maxwell-equations? Currents are generated by magnetic flux change. If magnetic flux can penetrate the conductive surface, then it's change can generate currents in a piece of conductor located inside the hollow.
We proved yet only that electrostatic field can't enter into the hollow. The proof is based on the fact that the surface of the conductive shell is equipotent. But what is the situation, when the external field changes in time? Then the potential of the shell evidently changes in time and in a given time instance varies point by point on the shell.
Can't it be that sudden changes of the electromagnetic field can penetrate the conducting surface? If yes, then how depends the transmission on the frequency? What is the mathematical description of this phenomenon?

 

[Astronuc]

Give me some time to read the thread and respond.

Electric currents are just moving charges, and the currents are induced by an electric potential (emf or voltage), a time varying magnetic field, or by a conductor containing free charges moving in a static (constant) magentic field.

 

[pervect]

OK, but how follows this from the Maxwell-equations? Currents are generated by magnetic flux change. If magnetic flux can penetrate the conductive surface, then it's change can generate currents in a piece of conductor located inside the hollow.
We proved yet only that electrostatic field can't enter into the hollow. The proof is based on the fact that the surface of the conductive shell is equipotent. But what is the situation, when the external field changes in time? Then the potential of the shell evidently changes in time and in a given time instance varies point by point on the shell.
Can't it be that sudden changes of the electromagnetic field can penetrate the conducting surface? If yes, then how depends the transmission on the frequency? What is the mathematical description of this phenomenon?

Faraday cages made out of conductors with finite resistance won't be effective against low frequency magnetic fields. (Superconducting faraday cages would be, of course).

I was surprised at how little information was available for the effectiveness of Faraday cages at power line frequencies (50-60 hz).

Combining info from the published specs of Faraday cages that I found

http://www.hollandshielding.be/prefab.htm

plus the skin depth of copper at 60 hz being 8 mm

http://en.wikipedia.org/wiki/Skin_effect

I don't think copper is going to be very effective for shielding 60 hz magnetic fields. (They'd work quite well for 60 hz electric fields, though).

Mu metal shielding appears to be the best way to shield against low frequency magnetic fields - superconductors would be even better if the expense is warranted, I suppose.

 

[leright]

Well trivially, it's Gauss's law. Inside the hollow conductor there is not charge, so the enclosed charge is zero, so the electric field is zero everywhere.

Now more directly, consider the most trivial case of the center of a hollow sphere, with 'uniform' charge on the surface. For each charge, there is an equal charge diametrically opposed, and therefore at the center the electrical fields (vectors) are equal and opposite, so they cancel.

Now, consider any point, off-center. One cannot apply the opposite point charge, but rather one must consider opposing surfaces, dA, which would have charges $\sigma_1\,dA_1$ and $\sigma_2\,dA_2$. Now think if two cones with vertices touching (and having same solid angles) and colinear (parallel) axes, with heights $r_1$ and $r_2$. The E from one is just $\sigma_1\,dA_1$/$r_1^2$ and the other is $\sigma_2\,dA_2$/$r_2^2$, but realize that $dA_i$ is proportional to $r_i^2\,d\Omega_i$, where $d\Omega$ is the solid angle enveloped by cones and subtended by $dA_i$.

So E_i is proportional to 1/$r_i^2$, and $dA_i$ is proportional to $r_i^2$, and the term cancel which then leaves equal charges ($\sigma\,d\Omega$) opposing each other, and therefore the electric fields cancel, i.e. $\vec{E}\,=\,0$.

Voila!

Gauss's law explains that the divergence of the e-field is zero when there is no charge enclosed by a closed surface, but just because the divergence is zero doesn't necessarily mean there's no e-field. An e-field can exist in the closed surface even if all flux entering also leaves. I think the explanation of the faraday cage is somewhere other than Gauss's law.

 

[mma]

I'd like to restrict our investigations to ideal conductors, i.e, having zero resistance and massless charge carriers.

The logic of our solution for static case was the following.

• The charges on the conducting surface will arrange such that they feel no force on itself (otherwise they can't be at rest).
• This arrangement corresponds to an equipotential surface
• In the inner region bounded by a closed equipotential surface, the potential is constant

In the dynamic case we have to find the analogue statements

• The charges on the conducting surface will move such that they ? (invoking Lenz's law?)
• This movements corresponds to an ?
• In the inner region bounded by a closed ? surface, ?

 

[leright]

No, the other solution that I provided ends up being a surface integral. If one makes $d\Omega$ very small and integrates the E field over all $\Omega$, one discovers that the entire integral vanishes, for any arbitrary surface enclosing a volume. All the example shows is that there is an equal charge on one side of the volume opposing the other charge, and the net E field is zero.

For non-spherical surfaces, and particularly nonsymmetric surfaces/volumes, the integrals can be more complicated.

What you're saying is that if you enclose a chargeless volume then there cannot be an E-field there. This is false.

There is a difference between flux and net outward flux. It is true that a sphere enclosing chargeless volume will have no net outward flux. However, there can still be an E-field, correct? An e-field entering one side of the sphere and exiting the other side can still exert a force on a small test charge placed in the sphere, correct? Just because there is no net outward flux doesn't necessarily mean that there is no e-field that can do work on a test charge.

 

[pervect]

I'd like to restrict our investigations to ideal conductors, i.e, having zero resistance and massless charge carriers.

The logic of our solution for static case was the following.

• The charges on the conducting surface will arrange such that they feel no force on itself (otherwise they can't be at rest).
• This arrangement corresponds to an equipotential surface
• In the inner region bounded by a closed equipotential surface, the potential is constant

In the dynamic case we have to find the analogue statements

• The charges on the conducting surface will move such that they ? (invoking Lenz's law?)
• This movements corresponds to an ?
• In the inner region bounded by a closed ? surface, ?

Zero resistance simplifies things.

Right off the bat, we can say:

1) in the electrostatic case, electric fields in the interior will be zero. This is because the voltage of the enclosing faraday cage will be constant everywhere on the surface. You can have fields inside the cage due to charges inside the cage, but E=0 will be the solution with no internal charges.

2) In the quasi-static electrostatic case, the voltage on the surface will be a slowly-varying constant. This will also give zero fields inside the sphere.

Circuit-wise, we can consider the situation to look like the following


(voltage source)---C1-----(faraday cage)----C2-----(ground)

Some capacitance C1 connects the faraday cage to some voltage source which generates a signal from the environment. Some capacitance C2 connects the Faraday cage to ground.

2a) The non-static case where the field varies rapidly needs some more thought on my part, this may be what the OP is interested in.

3) In the magnetostatic case, the flux through a loop of wire cannot change in a wire of zero resistance. Thus the magnetic field will not change. The case of a solid cage is equivalent to the case of a screen made out of small loops.

In an actual superconductor, not only will the magnetic flux through a superconducting loop be constant, it will always be zero (the Meisner effect). The superconductor will expel any existing flux when it becomes a superconductor.

3) For electromagnetic waves, any incoming wave will be totally reflected with an ideal conductor, and no wave will be transmitted.

 

[mma]

2a) The non-static case where the field varies rapidly needs some more thought on my part, this may be what the OP is interested in.

Yes, the point would be this.

However, the picture is complete only together with the overview you have provided here.