# Induced electric fields

1. Jul 7, 2014

### biomagnetico

Hi all,

I am looking into induced electric fields by changing B fields inside layers of dielectric. For example, if a block of several layers of a dielectric material is placed inside a capacitor plate, the E fields inside each layer is shielded depending on the relative permitivitty.

Question: what is the E field inside the same block if now it is placed inside a solenoid that has a changing B field thus establishing an induced field inside the dielectric block?

In other words, does the induced E field inside the block is the same all through out the different layers of dielectric (by Faraday's law) or does it scale depending on the relative permitivity of each layer?

2. Jul 7, 2014

3. Jul 7, 2014

### biomagnetico

Thanks, but it does not help.
Are you suggesting that induced E field is independent of dielectric?

Thanks!

4. Jul 7, 2014

### WannabeNewton

In the absence of electric material, Faraday's law can be written $\vec{\nabla}\times \vec{E} = -\partial_t B$. In the presence of electric material, such as dielectrics, Maxwell's equations equations can be preserved in the exact same form and manipulated in exactly the same way as usual if we replace the electric and magnetic fields appearing in certain equations with the axillary fields $\vec{D} = \epsilon_0\vec{E} + \vec{P}$ ($\vec{P}$ is the polarization per unit volume) and $\vec{H} = \frac{1}{\mu_0}\vec{B} - \vec{M}$ ($\vec{M}$ is the magnetization or magnetic moment per unit volume). Now this replacement is only for some of the equations, not all. It turns out that no replacements need to be made for Faraday's law; even in the presence of a dielectric, Faraday's law remains the equation $\vec{\nabla}\times \vec{E} = -\partial_t B$ for the actual rotational electric field $\vec{E}$ (as opposed to the auxiliary field $\vec{D}$) even in the presence of a dielectric or layers of dielectrics.

5. Jul 7, 2014

### biomagnetico

Thank you for the answer. So the induced E field is the same throughout the layers. How can the material differentiate whether the field is induced (d/dt B) or established by a capacitor?

In other words, the induced E field is on a different dimension, or level, as the E field from actual charges?

6. Jul 7, 2014

### WannabeNewton

The field associated with the capacitor is of course affected by the dielectric layers and in fact Gauss' law will instead happen to read $\vec{\nabla}\cdot \vec{D} = 4\pi \rho_f$ for the axillary field $\vec{D}$ defined above; here $\rho_f$ denotes the density of free charge in the system (which includes all the charges in the system that aren't the induced bound charges in the electric material from the polarization density $\vec{P}$). However another way to differentiate them is to simply note that the field generated by the charges on the capacitor, which is the field associated with Gauss's law, is irrotational meaning $\vec{\nabla} \times \vec{E}_G = 0$ where I've included the subscript just to indicate this is the electric field that comes straight from Gauss's law. The field associated with Faraday's law and induced by a time-varying magnetic field is of course rotational since $\vec{\nabla} \times \vec{E}_F = -\partial_t \vec{B}$.

7. Jul 7, 2014

### vanhees71

It is quite misleading to say "a time-varying magnetic fields induces an electric field" due to Faraday's Law. It is as well very misleading to interpret the time derivative of the electric field as a kind of current in the Ampere-Maxwell Law and interpret this "displacement current" as a kind of additional source of the magnetic field.

The correct interpretation of Maxwell's equations tells you that the sources of the electromagnetic field (with electric and magnetic components) are the charge and current density distributions. There is only one electromagnetic field, with the 6 field-degrees of freedom we usually write as $\vec{E}$ and $\vec{B}$ in the three-dimensional notation, which is indeed best for practical purposes, while for fundamental purposes the manifest covariant notation in Minkowski space-time is the most appropriate.

So far I've described electromagnetism in terms of the very fundamental ingredients. For macroscopic matter, it is impossible to work in this detail, and it is also totally unnecessary. That's why one uses the methods of statistical physics to describe appropriate macroscopic quantities (as you describe a gas in terms of fluid mechanics rather than as a complicated set of $10^{26}$ coupled equations of motion for the coordinates and momenta of all particles the gas is made of).

This leads in certain approximations (the socalled linear-responsse theory) to the known equations of macroscopic electrodynamics with the fundamental fields $\vec{E}$ and $\vec{B}$, which appear unchanged from the coarse-graining procedure of the homogeneous Maxwell equations, which read (in Heaviside-Lorentz units)
$$\vec{\nabla} \times \vec{E}+\frac{1}{c} \frac{\partial \vec{B}}{\partial t}=0, \quad \vec{\nabla} \cdot \vec{B}=0,$$
while the inhomogeneous equations get modified, because you only like to treat explicitly the motion of the charge and current distributions you bring additionally into the game, while the redistribution of the charges and currents of the matter consituents from equilibrium due to the electromagnetic field due to these additional charges, is treated via a coarse-graining procedure by the polarization and magnetization as explained by WannabeNewton.

The mathematical technique for this coarse-graining procedure is the multipole expansion of charge distributions, stopping right after the dipole approximation. Then you get
$$\vec{\nabla} \cdot \vec{D}=\rho_{\text{ext}}, \quad \vec{\nabla} \times \vec{H}-\frac{1}{c} \partial_t \vec{D}=\frac{1}{c} \vec{j}_{\text{ext}}.$$
In the linear-response approximation you make the ansatz that $(\vec{D},\vec{H})$ is a linear (and causal!) functional of $(\vec{E},\vec{B})$. In the most simple approximation of a homogeneous and isotropic medium you get just three constitutive functions, the permittivity, permeability, and conductivity of the material in question. These are usually functions of frequency in Fourier space (sometimes also of wave number, if there is spatial dispersion, like in plasmas) and temperature and other thermodynamic macroscopic quantities describing the medium.

For some materials it becomes even more complicated, e.g., for crystals you have not simply functions but tensors, describing phenomena like berefringence. In other cases you have considerable "memory effects" (non-Markovian behavior in the sense of kinetic theory). The most common example are ferro magnets with a hysteresis effect that can only be fully understood when refining the model for the medium with a quantum theoretical description.

8. Jul 7, 2014

### biomagnetico

Thanks, it definitely does not help --- it is not on the topic.

9. Jul 7, 2014

### Simon Bridge

I am suggesting that you apply Maxwel's equations to your problem - it won't help unless you actually do this.
I see no evidence that you have.
If you are not prepared to do this, then there is not much we can do to help you.

You could also consider what happens to light passing through different dielectrics - here you have E and B fields inducing each other - you can consider how the amplitudes of the waves are affected.

As for how material can tell the difference between an E field induced by an applied time-varying B field (Vanhees' correct and very relevant redirection notwithstanding) and an E-field applied to the same dielectric - you need to delve further into how E and B fields are related in those two situations.

Notice that everyone is basically trying to get you to come to a new understanding through your own efforts. That is because it won't do you any good for us to just tell you.

10. Jul 8, 2014

### biomagnetico

Thanks so much.

I have solved the problem with a layered dielectric is placed in between 2 plates. I believe I understand how the electric field is distributed and shielded according to each relative permittivity.

I have problems visualizing the concept with the dielectric block is placed inside a solenoid with a time varying field. According to Faraday's law, the induced field E is independent of the material. Therefore the induced field in air and in the dielectric layers is the same. I have a problem believing this.

If we compute the displacement field D inside each material (air, dielectric), and assuming the same E field, we obtain different D fields due to each relative permittivity.

You bring a good point about E and B field propagation in dielectric medium and how amplitude of E is modulated by relative permittivity.

I do not see how this follows in my example directly from Faraday's Law.

I am constantly thinking of the problem, so I do put effort on it. I dont want an answer, I am trying to understand it conceptually so I can apply a principle.

11. Jul 8, 2014

### vanhees71

I'm sorry, but I have to insist that this is physically not really correct and didactically misleading. You cannot say that E and B fields are inducing each other; neither in free space, nor in a dielectric.

The electromagnetic field is propagating in free space or through dielectric (i.e., non-conducting) matter as a whole. In a given (inertial) frame of reference it consists of 6 components, none of which induce each other in a causal sense.

Let's take the free Maxwell equations as the most simple example,
$$\vec{\nabla} \times \vec{E}+\frac{1}{c} \partial_t \vec{B}=0, \quad \vec{\nabla} \cdot \vec{B}=0,$$
$$\vec{\nabla} \times \vec{B} - \frac{1}{c} \partial_t \vec{E}=\frac{1}{c} \vec{j}, \quad \vec{\nabla} \cdot \vec{E}=\rho.$$
The usual way to solve these equations is to use the first two equations (the homogeneous Maxwell equations) to introduce the four-vector potential (or in the 1+3-notation a scalar and a vector potential in the 3D sense of Euclidean configuration space) and then deriving equations of motion for these potentials from the two inhomogeneous equations.

The didactical difficulty with this approach, however, is that you have to deal with gauge invariance, and that the potentials are gauge dependent and thus the causality structure of the entire concept is pretty subtle. Only in the Lorenz gauge the potentials are entirely the retarded solutions. In other gauges, e.g., Coulomb gauge, the potential can contain instantaneous pieces etc.

Of course, the physical em. field is represented by the field-strengths components, $\vec{E}$ and $\vec{B}$, and they are uniquely defined via apprpriate boundary conditions guaranteeing causality to be retarded solutions.

In classical electrodynamics we can work with the field strengths only. The idea is to separate the electric and magnetic field components, making use of Maxwell's equations. E.g., take the curl of the first inhomogeneous equation:
$$\vec{\nabla} \times (\vec{\nabla} \times \vec{B}) -\frac{1}{c} \frac{\partial}{\partial t} (\vec{\nabla} \times \vec{E})= \frac{1}{c} \vec{\nabla} \times \vec{j}.$$
We work in Cartesian coordinates from now on. Then we can write
$$\vec{\nabla} \times (\vec{\nabla} \times \vec{B}) =\vec{\nabla} (\vec{\nabla} \cdot \vec{B})-\Delta \vec{B}.$$
The first term vanishes due to the 2nd homogeneous equation. Using also the 1.st homogeneous equation, you find
$$\frac{1}{c^2} \partial_t^2 \vec{B}-\Delta \vec{B}=\frac{1}{c} \vec{\nabla} \times \vec{j}.$$
Here, it becomes immediately clear that the source of the magnetic field is the curl of the current density, and since $\vec{B}$ is a physically observable field, it must be causally connected to this source, i.e., the only physically sensible solution is the retarded solution, i.e.,
$$\vec{B}(t,\vec{x})=\frac{1}{4 \pi c} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x}' \; \left ( \frac{\vec{\nabla}' \times \vec{j}(t',\vec{x}')}{|\vec{x}-\vec{x}'|} \right )_{t'=t-|\vec{x}-\vec{x}'|/c}.$$
In the same way one can also eliminate the magnetic field components by taking the curl of the first homogeneous Maxwell equation (Faraday's Law)
$$\vec{\nabla} \times (\vec{\nabla} \times \vec{E})+\frac{1}{c} \frac{\partial}{\partial t} (\vec{\nabla} \times \vec{B})=0.$$
Again in Cartesian coordinates we have
$$\vec{\nabla} (\vec{\nabla} \cdot \vec{E})-\Delta \vec{E} + \frac{1}{c} \frac{\partial}{\partial t} (\vec{\nabla} \times \vec{B})=0.$$
From the 2nd inhomogeneous Maxwell equation (Gauß's Law), used on the first term, and the first inhomogeneous Maxwell equation (Ampere-Maxwell Law) on the third term we get after some rearrangement of terms
$$\frac{1}{c^2} \partial_t^2 \vec{E}-\Delta \vec{E}=-\vec{\nabla} \rho -1/c^2 \partial_t \vec{j},$$
which has the retarded solution
$$\vec{E}(t,\vec{x})=-\frac{1}{4 \pi} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x}' \; \left (\frac{\vec{\nabla}' \rho(t',\vec{x}') + \partial_t' \vec{j}(t',\vec{x}')}{|\vec{x}-\vec{x}'|} \right )_{t'=t-|\vec{x}-\vec{x}'|/c}.$$
This form of the retarded solutions of the Maxwell equations is known as "Jefimenko's equations" although they have been first derived already mid of the 19th century by Ludwig Lorenz.

Of course, you can rewrite the solutions in some way such as to look as if, say the magnetic field components are somehow "induced" by the electric field components. However, as you'll immediately see, the corresponding equation is not of local causal form as the Jefimenko equations, which in turn only contain the charge and current densities as sources of the em. field. Thus the much more adequate interpretation of Maxwell's equations is that the em. field is caused (or "induced" if you wish to use this word) by the charge and current density.

Of course, the electromagnetic field also propagates in charge-current-free space or in electrically neutral media in form of waves, but also the free Maxwell equations do not admit a rewriting such as some of its components are causally and locally connected with other of its components. The phenomenon of em.-field propagation must be simply seen as a phenomenon of the electromagnetic field as a whole. It follows as solutions of the homogeneous wave equations, which you have to add anyway to the Jefimenko solutions of the inhomogeneous one to fulfill boundary conditions appropriate to your problem. The Jefimenko solutions describe the situation where em. waves are created due to charge-current distributions in some (usually finite) part of space. The boundary condition then of course is that at all times the em. field must vanish at spatial infinity, and that's precisely the case for the Jefimenko equations. That one has to use the retarded and not the advanced (or some superposition of both) is due to the causality constraint that there should no effect before the cause, i.e., at each point in space time the electromagnetic field must be expressible in terms of quantities at this time or earlier times but not at later times. The only solution then is the purely retarded one as given in the Jefimenko solutions.

You can repeat all this, of course, for the case of wave propagation in dielectrics. Only the solutions become somwhat more complicated, because the wave equation is modified due to the dielectric as it must be, because you must describe the non-trivial phenomenon of dispersion (i.e., refraction) of the electromagnetic wave in the medium. For an execellent treatment of the classical dispersion theory for usual dielectrics, see

A. Sommerfeld, Lectures on Theoretical Physics IV (Optics)