Remark: This post is somewhat controversial. The trouble is that electrostatics and electricity are not well explained nor understood by most authors. The properties of Faraday cages is one such example, as we'll describe below.
We were brought to this subject reviewing Prof. AKT Assis' book with JA Hernandes "The Electric Force of a Current" pdf available from Prof. Assis' website. The Faraday cage arises in connection with a study of an experiment proposed by R. Sansbury related to "anomalous" electric forces. Assis remarks that Faraday cages have surprising properties which are elementary, but not well known. This was the beginning of our study.
Hollow Conductors:
Here is the setting. We imagine a solid conductor material, like a hollow spherical shell, where the width of the shell is actually rather thin. So the conductor is very much hollow.
To say the shell is a conductor, rather than an insulator, means the shell is made of metal or aluminum, something with abundance of free mobile electrons. By contrast a conductor is something made of glass or ceramic like porcelain, or potentially rubber or oil or plastics. We view an insulator as being made of lattice material which strongly binds the electrons in the lattice structure, and there is no mobility.
So we have a conductor $C$. It has a volume and it has a surface. Saying the conductor $C$ is hollow means $C$ has an interior volume. Topologically $C$ divides the ambient space ${\bf{R}}^3$ into two volumes ${\bf{R}}^3 - C = V_i \coprod V_o$. We are considering $V_i, V_o$ as consisting of the inner and outer volumes bounded by $C$.
Remark.
The coproduct symbol "$\coprod$" emphasizes that the volumes are disjoint.
If the conductor $C$ was not hollow, but rather a solid torus for example, then it's not necessary that $C$ divide the ambient space into two "inner" and "outer" components.
If the conductor $C$ really is a volume than the boundary intersections $\partial V_i$ and $\partial V_o$ are disjoint and partition the topological boundary of the conductor $C$. For example a hollow sphere actually has two boundary components, namely the inner and outer surfaces, which are disjoint, and which are the respective boundaries of the inner and outer volumes.
Faraday Cages
We begin with a fundamental theorem of Poisson which is related to the "method of images". We quote from Weber, writing on this theorem of Poisson (1812):
- "When arbitrary electric forces act upon a conductor of arbitrary form from the outside, a distribution of free electricity on the surface of the conductor is always possible -- but only one of them -- for which the electric forces that originate from that distribution of free electricity will likewise be in equilibrium with the electric forces at all points of the interior of the conductor that act from the outside."
Notice the occurence of terms like "from the outside", "on the surface", "of the interior". Moreover Poisson seems to restrict his theorem to the case of "arbitrary electric forces acting from the outside". But in practice we can imagine the external forces have non electric origin. For example, Weber applies Poisson's theorem to steady currents where the external force arises from the electrochemical force of a battery (Voltaic pile).
Here's the formal statement of Poisson's theorem:
- "If we have a conductor volume $C$, and a net external force $F_{ext}$ acting on $C$, then there exists a unique surface charge distribution $\sigma$ on the boundary $\partial C$ such that $$F_{ext}(x)+F_\sigma(x) =0~~~\text{for every}~~x\in C."$$
Remarks.
The main issue is that the inside of $C$ and the interior volume bounded by $C$ are frequently confused, especially in the case when the conductor $C$ is a very thin volume, e.g. thin hollow spherical shell $"O"$.
Poisson claims that the net force $F_{net}(x)=0$ vanishes for $x\in C$ inside the conductor. Confusion arises when the inside of the conductor is mistakenly identified with the interior volume bounded by $C$. Thus Poisson needs clarify whether $F_{net}(x)=0$ for all $x\in C$ or for all $x\in V_i$. We think Poisson proves the case for all $x\in C$.
We find most questions about Faraday cages assume that the net force vanishes for all points in the interior volume $V_i$. And this confused us for some time, since it's evidently easy to mistake the "inside of $C$" with the outer volume $V_i$ which is bounded by $C$. But strictly speaking: topology proves that the boundary $ \partial V_i$ is not equal to $\partial C$ but only a subset of $\partial C$.
The most important aspect of Poisson's theorem is the proof via minimum energy principle, or least action principle, as developed by Lord Kelvin (William Thompson) and George Green (1830s--1850s). This is elaborated by Hammond (see below). And here enters important divergence theorems which need be strictly applied to the correct topological boundaries and volumes.
The surface charge $\sigma$ is obtained via the normal derivative of the potential $\phi$ along the boundary, namely $\frac{\partial \phi}{\partial n}$, which is by definition equal to $\nabla \phi \cdot {\bf{n}}~ dS$ where $\bf{n}$ is the "outer" normal of the surface.
Hammond's Energy Methods and Green-Thompson Theorems:
The equation $F_{ext}+F_\sigma = 0$ is interesting, and we've been reviewing Hammond's textbook "Energy Methods in Electrostatics", and especially Ch.4 on Energy Theorems.
He reviews George Green's essays on mathematical analysis of electricity papers, which are actually very readable and well written (from 1830s!). We especially like how Green addresses the problem of evaluating divergence theorems when the differentials diverge to infinity. His attention to singular values is very much interesting!
The analysis begins with Green's first formula, which is the divergence theorem combined with a product formula. For the conductor $C$ we obtain the identity
[eq1] $$\int_C \phi .\Delta \phi~ d\text{vol}+ \int_} \phi .\sigma~ dS = \int_C \nabla\phi\cdot\nabla \phi ~d\text{vol}.$$
and for the volumes $V=V_i$ say, we have:
[eq2] $$\int_V \phi .\Delta \phi~ d\text{vol}+ \int_} \phi .\sigma~ dS = \int_V \nabla\phi\cdot\nabla \phi ~d\text{vol}.$$
If there is a net external force $F_{ext}$, then potentially that force is nonzero throughout $C$ and $V_i$, $V_o$.
Poisson's theorem argues that within a conductive medium, e.g. $C$, there is no excess free charge density in the interior of $C$, which is not $V_i$ but $C$. This is how we obtain harmonicity $\Delta \phi=0$ throughout $C$.
Moreover the conductivity of $C$ is extremely different from conductivity of $V_i$. Actually $V_i$ is not conductive, being essentially considered a vacuum, or at least until a test charge $q$ is introduced into $V_i$.
Hammond calls the right hand term the "field energy" of the system. He emphasizes that the inclusion of the surface charge $\sigma = \epsilon_0 \frac{\partial \phi}{\partial n}$ in the left-hand side integral is necessary for the total energy of the system to be uniquely defined. This total energy is represented by the right hand integral involving $|\nabla \phi|^2 = \nabla \phi \cdot \nabla \phi$.
The conductivity factor $\epsilon$ is obviously dependant on the material medium of the specific volume. In the case of Faraday cages, this quantity is vastly different for the empty interior of the volume, and for the conductive surface of the "cage". It's important not to ignore this necessary factor, since evidently $\epsilon_C \neq \epsilon_V$.
The quantity $\phi. \nabla \phi$ is called the volume energy density by Hammond. The above Green formula implies it has units of energy density (hence integrating we obtain an energy quantity). We prefer to interpret $(\phi.\Delta \phi)$ as the assembly work required to build the system. This reflects the work that was necessary to build (or assemble) the point charges defining the potential $\phi$. This amount of work is represented in $\int_V \phi \rho$. So there is energy inherent in the system, as represented by this quantity. Apparently William Thompson Lord Kelvin wrote something like this in his diary in 8 April 1845.
Thompson made the further observation that the Green identities represent an equilibrium equation, or ground state. This is treated in Hammond's book.
So what are we saying, that Faraday cages don't work? That you shouldn't put your face close to the microwave?
No. Maybe.
We are saying that there is an electric field present in the interior volume of a Faraday cage. I.e. $\nabla \phi \neq 0$ in the interior volume $V_i$.
And this is contrary to the expectations of Faraday, Feynman, etc.. Perhaps there is important distinction between thin cages used in experiments and the solid enclosed conductor which keeps the effect negligible, i.e. alot of net cancellation.
However the arguments are not persuasive. Truly there's no rigour, so we think we are well permitted to question the validity of the result.
The argument is usually made via Gauss' law, except Gauss Law does not determine the electric field, i.e. $\nabla \phi$. Instead Gauss Law determines $\nabla \cdot (\nabla \phi)=\Delta \phi$ in terms of the electric charge density $\rho$. So the potentials within $V_i$ will be harmonic.
Poisson's theorem says the net force inside the conductor $C$ proper is zero, and the electric force is defined by a surface density $\sigma =\epsilon \partial \phi / \partial n$. If there is zero external force, then and only then does Poisson predict zero electric force in the interior of the conductor.
The above paragraph demonstrates a readily made mistake, a misreading of Poisson's theorem as stating that the electric force is zero. That is not what Poisson proves! He proves that given an external force $F_{ext}$ there will be a rearrangement of the free electrons inducing a surface charge distribution $\sigma$ such that the electric force $F_\sigma$ exactly equilibriates with the external force to achieve zero net force: $$0=F_{net} = F_{ext} + F_\sigma,$$ or equivalently $$F_\sigma = -F_{ext}.$$ Therefore it's evident, but typically overlooked, that there is only zero electric force if there is zero external force, i.e. isolated system.
These are trivial remarks, but need to be said.
For example, the typical confusion believes that there is zero electric force inside the conductor, and therefore the potential $\phi$ is constant throughout the conductor. Moreover the surface charge distribution is obtained by the normal derivative $\partial \phi / \partial n$. But if $\phi$ is constant throughout the interior, then isn't the normal derivative zero? I.e. there is zero surface charge density? This is the contradiction that is confusing authors.
..."But What About Experiments?"
E.g. Benjamin Franklin 1750s? Or Faraday's own personal cage?
But what about the experiments, why do they apparently work? Why are we protected from outside lightning when we are sitting in our car?
The above screenshots are from "Prelude to Power" an amusing dramatization of Faraday's discoveries.
Katie Loves Physics Explaining Some History of Faraday Cage. Contains useful historical references.
Physics Forum Question with Very Unsatisfactory Answers. See the erroneous answer invoking Gauss' law. There is no way to deduce a vanishing electric field from a locally vanishing charge density.
https://physics.stackexchange.com/questions/729232/field-inside-hollow-conducting-sphere?rq=1
https://physics.stackexchange.com/questions/617489/electric-field-inside-a-cavity-faraday-cage
https://physics.stackexchange.com/questions/192266/electric-field-inside-hollow-conducting-bodies
There's alot of different answers, but always ending with a nonexplanation, simply the claim that the resultant electric fields cancel to zero.
There is also appeal to Gauss' theorem, but this describes only the divergence of the gradient of the potential, and not the gradient itself.
It's true that there is distinction between the empty region, where the electric field is zero because there is no charge distribution in the conductor in the absence of an external field.
If one brings a test charge $+q$ near the conductor, then the free electricity inside the conductor will rearrange into the minimal position, generating a net zero force in the interior of the conductor. But there again will be a net electric field in the exterior of the conductor! This can be argued by the method of images. The test charge $+q$ induces a charge in the conductor equivalent formally to the effective field of an image charge $-q_i$. The net field is therefore the sum of the potentials generated by these point sources $+q$ and $-q_i$. And in this construction the conductor becomes an equipotential surface, but we also see that the electric field is nonzero within the interior volume of the conductor!
For example, they say that Faraday built a cubical room lined with metal sheets, like surrounding the walls of your house with aluminum foil, and exposed the room to electrical discharges from the outside volume. Faraday claims to have monitored the interior volume of the cube with an electrometer and detected no electric forces either at the boundary walls or anywhere in the interior of the room. This claim might appear to contradict our arguments above, but let's reserve our judgement and examine the situation more closely.
Mathematics of Faraday Cage?
Apparently the authors of [this paper] were shocked to find no readily available references or mathematical explanations of Faraday's cage. Consider the misleading and erroneous answers supplied in the above Physics Form link.
A sketch of the mathematics behind the cage is roughly: we are looking for harmonic functions $\phi$ on the conductor and its surroundings. The assumption is that $\phi$ is constant throughout the conductor and therefore $\nabla \phi=0$. And the gradient vanishing says there is no electric force.
So we can phrase everything in terms of harmonic functions. We are given a conductor $C$ and the space complement ${\bf{R}}^3-C$. So we are looking then for harmonic functions $\phi$ defined on the global space ${\bf{R}}^3$ satisfying the condition that $\nabla \phi|_C=0$. If the conductor is nontrivial volume, then doesn't the vanishing of $\nabla \phi$ imply $\phi$ is constant, and therefore vanishing everywhere?
Prof. Lewin's experiment does not seem convincing. He has a conductive ping pong. Are we expected to believe that if his pingpong touches the positively charged surface, then this positive charge will be imparted to the pingpong rod, and eventually ? This is only if the positive charges are mobile and attracted to the rod.
He touches one side of the box and it's negatively charged. He touches the other side of the box, and he sees that it's positively charged. (He's collecting these charges using the condutive ping pong, then discharging at an electrometer to see the polarity, positive or negative charge). But then Prof. Lewin places the pingpong in the volume bounded by the conductor can, and he moves the ping pong around and seems to indicate that he collects no free electric charges. The absence of charges is shown by the electrometer registering no charge. But what does this show? Not that there is zero net force in the interior, but rather that there are no free electrons!
Is this fair critique? There are no mobile positive ions which are collected by the pingpong and then deposited. Rather there's only mobile electrons. On the negative side, there is an excess of electrons which are possibly attracted to the pingpong. On the positive side there is an absence of electrons. When the pingpong conductor is placed in physical contact with the box, there is a flow of electrons from the pingpong conductor to the box.