Permittivity = EM conductivity?

[Saw]

I was wondering whether, making a comparison between the propagation of electric current through a conductor and the propagation of an electromagnetic wave through a dielectric, one could establish the following correspondences:

Ability of a material to facilitate the propagation of the “influence”:

- If the influence is the electric current and the material is a conductor, that is its CONDUCTIVITY.
- If the influence is an EM wave and the material is a dielectric (that is to say, an EM CONDUCTOR), that is its PERMITTIVITY, which could thus be labelled as EM CONDUCTIVITY.

Ability of a specific sample of the material in question to propagate the “influence”:

- If we talk about an electric current and a conductor, that is… CONDUCTANCE? Does this concept exist at all?
- If we speak about an EM wave traveling through a dielectric, that is its CAPACITANCE, which could also be labelled as… EM CONDUCTANCE.

I had this is mind and the PF’s entry on permittivity gave me a sort of confirmation. Certainly, the terms permittivity and capacitance may be more revealing in other contexts. But, if the context of the discussion is propagation of the EM wave, I would like to know if this is a correct didactic approach: whether one could say, “well, it will propagate better or worse depending on the EM conductivity of the material and the EM conductance of the object in question (based on its geometric characteristics), which is what you call, respectively, permittivity and capacitance”.

 

[DrDu]

Conductivity and permittivity in deed both are merely different descriptions of the same phenomenon. In the case that the charges which carry the current are bound (i.e. the response of an insulator on a alternating electric field), formulation in terms of permittivity allow to describe the response of the medium to an external field in a local way, something that is not clear when working directly with the current.

 

[tiny-tim]

Hi Saw!

I was wondering whether, making a comparison between the propagation of electric current through a conductor and the propagation of an electromagnetic wave through a dielectric, one could establish the following correspondences:

Ability of a material to facilitate the propagation of the “influence”:

- If the influence is the electric current and the material is a conductor, that is its CONDUCTIVITY.
- If the influence is an EM wave and the material is a dielectric (that is to say, an EM CONDUCTOR), that is its PERMITTIVITY, which could thus be labelled as EM CONDUCTIVITY.
…
I had this is mind and the PF’s entry on permittivity gave me a sort of confirmation.

Certainly, the terms permittivity and capacitance may be more revealing in other contexts. But, if the context of the discussion is propagation of the EM wave, I would like to know if this is a correct didactic approach: whether one could say, “well, it will propagate better or worse depending on the EM conductivity of the material and the EM conductance of the object in question (based on its geometric characteristics), which is what you call, respectively, permittivity and capacitance”.

Yes … R C and L have essentially similar effects in a circuit (if we regard reactance as similar to resistance ),

and there seems no reason not to apply the same concept to a general electromagnetic field.

In a circuit, the concept of inductance times length per cross-section area ("inductivity"?) is fairly natural … you could go into a shop and ask the electro-butcher to cut you off so many cm of inductor, just like a resistor, or a sausage!

I'm not sure whether capacitors come in lengths in the same way … if they do, then again we could talk of "capacitivity".

(And of course, conductivity is the inverse of resisitivity.)

And "inductivity" and "capacitivity" are just (totally unofficial) alternative names for permeability (H/m) and permittivity (F/m). ​

 

[Saw]

Thanks to both. I wonder though, now, whether the assimilation between permittivity and EM conductivity should not come with a caveat.

If I am not mistaken, electrical conductivity comes with a connotation of velocity. If the conductor is more conductive, the current is more intense, ie more charges cross a patch of the wire in a given time. But the more permittivity that a dielectric has, the more it slows down the EM wave (the index of refraction is higher).

Thus, the analogy would be valid in terms of transparency to energy flow versus absorption/dissipation. But not in terms of the time rate of the flow...

 

[kmarinas86]

In a circuit, the concept of inductance times length per cross-section area ("inductivity"?) is fairly natural … you could go into a shop and ask the electro-butcher to cut you off so many cm of inductor, just like a resistor, or a sausage! :biggrn:

I'm not sure whether capacitors come in lengths in the same way … if they do, then again we could talk of "capacitivity".

(And of course, conductivity is the inverse of resisitivity.)

And "inductivity" and "capacitivity" are just (totally unofficial) alternative names for permeability (H/m) and permittivity (F/m). ​

Would the "length" be that which is along the lines of flux, and the area that which cuts through the lines of flux? In the case for "inductivity", would we have to consider the length along the lines of magnetic flux as opposed to that of lines of electric flux?

 

[kmarinas86]

Permittivity and conductivity essentially differ by a factor of time (if you divide permittivity by conductivity) or frequency (if you divide conductivity by permittivity). I would like to know how the analogy between permittivity and "EM conductivity" can hold despite this apparent difference of units and what this difference has to do with the relationship between the two.

 

[DrDu]

Thanks to both. I wonder though, now, whether the assimilation between permittivity and EM conductivity should not come with a caveat.

If I am not mistaken, electrical conductivity comes with a connotation of velocity. If the conductor is more conductive, the current is more intense, ie more charges cross a patch of the wire in a given time. But the more permittivity that a dielectric has, the more it slows down the EM wave (the index of refraction is higher).

Thus, the analogy would be valid in terms of transparency to energy flow versus absorption/dissipation. But not in terms of the time rate of the flow...

Both the current and the polarization ( or the field D) increase with electric field E.
For an electromagnetic wave with frequency omega: (epsilon_r-1)epsilon_0=sigma/omega where epsilon_r and epsilon_0 are the relative permittivity and the dielectric constant of the vacuum, respectively, and sigma is the conductivity.

 

[kmarinas86]

Both the current and the polarization ( or the field D) increase with electric field E.
For an electromagnetic wave with frequency omega: (epsilon_r-1)epsilon_0=sigma/omega where epsilon_r and epsilon_0 are the relative permittivity and the dielectric constant of the vacuum, respectively, and sigma is the conductivity.

This appears to answer my second post in this thread, which is just previous to yours. So I wonder what happens when you have a high frequency AC circuit. Don't both factors become relevant at the same time? Probably together with "inductivity" as well?

 

[tiny-tim]

Thanks to both. I wonder though, now, whether the assimilation between permittivity and EM conductivity should not come with a caveat.

the pf library article isn't "assimilating" permittivity and conductivity, merely commenting on the names … each is a measurement (capacitance or conductance, respectively) times length per cross-section area!

Would the "length" be that which is along the lines of flux, and the area that which cuts through the lines of flux?

yes … any electric field can be replaced by imaginary tubes of capacitors following tubes of lines of the electric field, with the electric displacement (electric dipole moment) (charge times distance) between adjacent plates adjusted so that the electric displacement density (charge per area) D, measured in coulombs per square metre (C/m²), matches E/ε along the centre of each tube …

make the tubes narrow enough, and we can match the whole E field to any required degree of accuracy …

the length and area (for the "capacitivity") would be the length and area of a section of tube: in the same material, it would be the same for all tubes

In the case for "inductivity", would we have to consider the length along the lines of magnetic flux as opposed to that of lines of electric flux?

yes: in that case, instead of tubes of capacitors, each tube would be wound with a single current-carrying solenoid, and would follow tubes of the magnetic field

again, make the tubes narrow enough, and we can match the whole B field to any required degree of accuracy

 

[DrDu]

This appears to answer my second post in this thread, which is just previous to yours. So I wonder what happens when you have a high frequency AC circuit. Don't both factors become relevant at the same time? Probably together with "inductivity" as well?

Which both factors?

 

[kmarinas86]

Which both factors?

I meant permittivity and conductivity.

 

[DrDu]

I meant permittivity and conductivity.

Then, as they are mathematically equivalent, there is only one factor.

 

[Saw]

the pf library article isn't "assimilating" permittivity and conductivity, merely commenting on the names … each is a measurement (capacitance or conductance, respectively) times length per cross-section area!

I know, you (the PF library article) were not assimilating permittivity and conductivity.

You were just commenting that another name for permittivity could be capacitivity.

I was the one doing the assimilation, in my OP, between electric conductivity in a conductor (free electrons transmitting charge) and electro-magnetic capacitivity = permittivity in a dielectric (bound electrons transmitting not a charge but a charge oscillation).

Then I realized that there is a difference between capacitance and conductance and hence between capacitivity (= permittivity) and conductivity, with these words:

electrical conductivity comes with a connotation of velocity. If the conductor is more conductive, the current is more intense, ie more charges cross a patch of the wire in a given time. But the more permittivity that a dielectric has, the more it slows down the EM wave (the index of refraction is higher).

Thus, the analogy would be valid in terms of transparency to energy flow versus absorption/dissipation. But not in terms of the time rate of the flow...

kmarinas86 pointed in the same direction here:

Permittivity and conductivity essentially differ by a factor of time (if you divide permittivity by conductivity) or frequency (if you divide conductivity by permittivity).

but was probably not satisfied with my former explanation when saying:

I would like to know how the analogy between permittivity and "EM conductivity" can hold despite this apparent difference of units and what this difference has to do with the relationship between the two.

However, DrDu seems later to identify both things:

Then, as they are mathematically equivalent, there is only one factor.

I would like to narrow down the subject in the line of the OP and also what kmarinas86 asked above:

To what extent is permittivity the conductivity of an EM wave?

Please note that the question refers to how permittivity of the material contributes to the conduction of an EM wave, that is to say, “something” that causes a charge oscillation, not a simple charge attraction or repulsion.

If we did talk about a simple attraction or repulsion, it is my understanding that permittivity would play, precisely, the opposite role. The higher the permittivity, the weaker the force between two charges, the weaker the electric field, because the dipoles attenuate the electric field. Instead, if the traveling thing is an oscillation, then the dipoles transmit it. So permittivity plays for EM waves the role that conductivity plays for current, with the only difference that the electric conductivity favors the speed of the current while permittivity slows down the EM wave?

 

[tiny-tim]

I meant permittivity and conductivity.

Then, as they are mathematically equivalent, there is only one factor.

are you misreading that as "permittivity and capacitivity"?

I was the one doing the assimilation, in my OP, between electric conductivity in a conductor (free electrons transmitting charge) and electro-magnetic capacitivity = permittivity in a dielectric (bound electrons transmitting not a charge but a charge oscillation).

it isn't oscillation of the bound electrons, it's just (average) separation, of the electrons from their "bound" positive charges …

every "bound" pair is mini-micro-capacitor!

(capacitance is the ability to separate charge … the capacitance of a material tells us how far these "bound" pairs are separated, per volt)

Then I realized that there is a difference …

i'll reply more, later

 

[DrDu]

are you misreading that as "permittivity and capacitivity"?

No, what I am saying is that an electromagnetic wave inside a conductor (and also inside an isolator) can be described either (and equivalently) using conductivity or a complex dielectric constant.

 

[Dickfore]

If you write down the fourth Maxwell's equation for a material with a (relative) permitivity tensor $\hat{\epsilon}(\mathbf{x}, \omega)$, and a conductivity tensor $\hat{\sigma}(\mathbf{x}, \omega)$ in frequency domain, you have:

$$\epsilon_{i j k} \, \frac{\partial H_k}{\partial x_j} = -i \, \omega \, \epsilon_0 \, \epsilon_{i k}(\mathbf{x}, \omega) \, E_{k}(\mathbf{x}, \omega) + \sigma_{i k}(\mathbf{x}, \omega) \, E_{k}(\mathbf{x}, \omega)$$

It is easy to see that you can atribute the conductivity tensor as an imaginary part of a complex permitivity tensor:
$$\hat{\tilde{\epsilon}} \equiv \hat{\epsilon} + i \frac{\hat{\sigma}}{\epsilon_0 \, \omega}$$
or, attribute the permitivity tensor as an imaginary part of a complex conductivity tensor:
$$\hat{\tilde{\sigma}} \equiv \hat{\sigma} - i \, \omega \, \epsilon_0 \, \hat{\epsilon}$$
These are two complementary ways to combine the same physics into one unified form ($\hat{\tilde{\sigma}} = -i \, \omega \, \epsilon_0 \, \hat{\tilde{\epsilon}}$). The meaning of a complex response function is revealed by going back to time domain. It is responsible for a time shift of the effect w.r.t. the cause.

 

[DrDu]

Exactly, and if you allow for a non-local dependence of epsilon on x, (or equivalently a dependence on wavevector k) than you can even absorb the magnetic effects into the dielectric tensor.

 

[Saw]

it isn't oscillation of the bound electrons, it's just (average) separation, of the electrons from their "bound" positive charges …

every "bound" pair is mini-micro-capacitor!

(capacitance is the ability to separate charge … the capacitance of a material tells us how far these "bound" pairs are separated, per volt)

My understanding of permittivity is the ability of a material to polarize, in response to an applied field. I suppose that in the concept of “polarization”, the separation between the two poles may count, yes, so that… wider separation means more effective polarization? But is that the only relevant factor? For example, if polarization is by rotation, isn’t it harder to achieve in a viscous material?

In any case, I believe we are contemplating different physical situations:

- You seem to be thinking of a capacitor surrounded by conductors and a direct current. In that case, yes, there is no oscillation.

- But I am considering situations where there is oscillation. For example, if the conductors conduct an alternate current. Or if there are no conductors and the dielectric (e.g., glass) faces an electromagnetic wave (e.g., light). Here the dipoles do oscillate between one position and the opposite in response to the oscillations of the stimulus.

DrDu, Dickfore, your comments may contain the answer, but they are too advanced for me. I wonder if you could explain further.

 

[tiny-tim]

My understanding of permittivity is the ability of a material to polarize, in response to an applied field. I suppose that in the concept of “polarization”, the separation between the two poles may count, yes, so that… wider separation means more effective polarization? But is that the only relevant factor? For example, if polarization is by rotation, isn’t it harder to achieve in a viscous material?

here's a nice picture of the different forms of polarisation (ionic dipolar atomic and electronic), and their appropriate frequency ranges …

http://en.wikipedia.org/wiki/File:Dielectric_responses.svg

 

[tiny-tim]

No, what I am saying is that an electromagnetic wave inside a conductor (and also inside an isolator) can be described either (and equivalently) using conductivity or a complex dielectric constant.

ah, i see

in an RC circuit, with an AC supply, we're familiar with the impedance being Z = R + iX, with X being the reactance, proportional to 1/capacitance, and R of course being proportional to 1/conductance

so in that sense capacitance and conductance (or permittivity and conductivity) are intimately connected

(though, despite reading eg http://en.wikipedia.org/wiki/Electr...uctivity#Complex_resistivity_and_conductivity …

An alternative description of the response to alternating currents uses a real (but frequency-dependent) conductivity, along with a real permittivity. The larger the conductivity is, the more quickly the alternating-current signal is absorbed by the material (i.e., the more opaque the material is).

… i confess i don't immediately see how they combine directly into a useful complex number )

 

[Saw]

in an RC circuit, with an AC supply, we're familiar with the impedance being Z = R + iX, with X being the reactance, proportional to 1/capacitance, and R of course being proportional to 1/conductance

so in that sense capacitance and conductance (or permittivity and conductivity) are intimately connected

Well, yes, that is one of the situations where one can visualize this connection:

- An Alternating Current supply, a conductor and a capacitor.
- In the conductor, the AC flows (= there is organized motion of free electrons as driven by the voltage) to the extent that there is Conductance and it does not to the extent that there is Resistance (disorganized motion due to temperature).
- When the AC reaches the capacitor, the AC is transmitted, because the bound electrons polarize in one and the opposite direction, to the extent that there is Capacitance (which can thus be named as dielectric oscillation Conductance), or is not transmitted, to the extent that there is Capacitative Reactance (which could thus be called as dielectric oscillation Resistance).

The other situation, as commented, is an EM wave being transmitted through the same dielectric.

If there is connection at the level of a specimen of the material (Conductance versus Capacitance), there is also at the level of the materials (conductivity versus capacitivity = permittivity).

The question is still whether there is disconnection, to some extent.

It seems the analogy is valid in terms of transmission versus absorption (dissipation) of energy. Thus a conductor transmits electric energy, no matter whether the current is direct or alternating, in proportion to its Conductance. Likewise, a capacitor transmits oscillations (AC and EM waves) in proportion to its Capacitance = oscillating Conductance.

However, the analogy did not seem to be valid in terms of time rate of the flow... unless we are being told that it is, because the time element comes with the oscillation frequency...

… i confess i don't immediately see how they combine directly into a useful complex number )

Neither do I, also after reading all Wikipedia entries on related concepts.

 

[DrDu]

The response of a material to an applied field is due to the induced electric currents in the material.
At higher frequency it does not make sense to distinguish between the motion of bound charges and free charges, a distinction which is hard to make precise, anyhow (electronic wavefunctions extend over the whole body). Hence, outside (quasi-)electrostatics, one prefers to describe nowadays the behaviour of a material exclusively in terms of a complex permittivity.
Sometimes it may make sense to use an equivalent expression using conductivity, but the important word here is "equivalent".
Qualitatively, we have the following picture: Under the influence of the oscillating electric field, the electrons move sinusoidal around some center. At the point of maximal excursion the dipole moment density (or macroscopic polarization is maximal). On the other hand the current is maximal when the electrons pass through their equilibrium position. Hence the current is out of phase by 90 degree (yielding an "i" in frequency representation).
For fixed amplitude of oscillation, the maximal current increases with frequency nu or omega.
Hence Polarization P(omega)=(epsilon(omega)-epsilon_0)E(omega)=i j(omega)/omega =i sigma(omega)/omega E(omega), where j is the current density.

 

[Saw]

Well, for me this issue of the complex concept is most interesting, but new and I am trying to assimilate it, also with the aid of some parallel readings. So I find it hard to follow you, but for sure it is easier for you to follow me.

What I gather so far from your comments is that, given an AC (an oscillation), the ability of the material to transmit the same is given by a complex number, where both permittivity and conductivity are involved, one playing the role of real number, the other of imaginary number (they can swap roles).

But what I was trying to do is comparing that concept with the corresponding one when the stimulus is a DC, that is to say, "simple conductivity" (?).

If you had to make a comparison between the two things (the ability of a material to transmit DC versus its ability to transmit AC), which differences and similarities would you note?

(I leave out, for simplicity, the ssue of the transmission of an EM wave, which was the original question.)

 

[Dickfore]

For DC, you need to take $\omega = 0$.

 

[DrDu]

(I leave out, for simplicity, the ssue of the transmission of an EM wave, which was the original question.)

That's not necessarily a simplification and I was specifically referring to the AC case.
The problem is that the contribution of conductivity to the dielectric constant diverges as omega->0 (as it is proportional to 1/omega) . So in the static case, it is really better to treat conductivity and permittivity separately. Nevertheless due to that divergence, it is then usually sufficient to consider either only conductivity (in metals) or permittivity (in isolators), as the contribution of conductivity, if present, will completely dominate permittivity.

 

[Saw]

I have tried to put this formula

Hence Polarization P(omega)=(epsilon(omega)-epsilon_0)E(omega)=i j(omega)/omega =i sigma(omega)/omega E(omega), where j is the current density.

in Latex:

$P(\omega ) = [\varepsilon (\omega ) - {\varepsilon _o}]E(\omega ) = \frac{{ij(\omega )}}{\omega } = \frac{{i\sigma (\omega )}}{{\omega E(\omega )}}$

Is this right?

For DC, you need to take $\omega = 0$.

So in this case we get division by zero?

 

[PhilDSP]

It is easy to see that you can atribute the conductivity tensor as an imaginary part of a complex permitivity tensor:
$$\hat{\tilde{\epsilon}} \equiv \hat{\epsilon} + i \frac{\hat{\sigma}}{\epsilon_0 \, \omega}$$
or, attribute the permitivity tensor as an imaginary part of a complex conductivity tensor:
$$\hat{\tilde{\sigma}} \equiv \hat{\sigma} - i \, \omega \, \epsilon_0 \, \hat{\epsilon}$$
These are two complementary ways to combine the same physics into one unified form ($\hat{\tilde{\sigma}} = -i \, \omega \, \epsilon_0 \, \hat{\tilde{\epsilon}}$). The meaning of a complex response function is revealed by going back to time domain. It is responsible for a time shift of the effect w.r.t. the cause.

Very nice presentation! Usually when i pops up signaling a half PI radian phase shift it's because one of the terms is defined as a first order derivative of something else. Looks like it's time to hit the books again on this to find out what that underlying term might be.

 

[DrDu]

Very nice presentation! Usually when i pops up signaling a half PI radian phase shift it's because one of the terms is defined as a first order derivative of something else. Looks like it's time to hit the books again on this to find out what that underlying term might be.

What is the phase relation between the dipole moment and the current for an oscillating dipole?

 

[DrDu]

I have tried to put this formula



in Latex:

$P(\omega ) = [\varepsilon (\omega ) - {\varepsilon _o}]E(\omega ) = \frac{{ij(\omega )}}{\omega } = \frac{{i\sigma (\omega )}}{{\omega E(\omega )}}$

Is this right?



So in this case we get division by zero?

The E in the last term on the right should be in the numerator, not denominator.
Yes, we get division by 0.

 

[Saw]

The E in the last term on the right should be in the numerator, not denominator.

Thanks. So after this correction we have:

$P(\omega ) = [\varepsilon (\omega ) - {\varepsilon _o}]E(\omega ) = \frac{{ij(\omega )}}{\omega } = \frac{{i\sigma (\omega )E(\omega )}}{\omega }$

and also:

$\varepsilon (\omega ) = {\varepsilon _o} + \frac{{i\sigma (\omega )}}{\omega }$

Is this the same as what Dickfore had said before?

It is easy to see that you can atribute the conductivity tensor as an imaginary part of a complex permitivity tensor:
$$\hat{\tilde{\epsilon}} \equiv \hat{\epsilon} + i \frac{\hat{\sigma}}{\epsilon_0 \, \omega}$$

 

[Dickfore]

So in this case we get division by zero?

Provided that $\hat{\sigma}(\omega) \neq o(\omega), \ \omega \rightarrow 0$. This is true for conductors, but not for insulators (dielectrics).

EDIT:
Remember, we are working wIth the Fourier transforms of these quantities in the frequency domain. This simply tells us that there ought to be a pole of the dielectric response function at $\omega = 0$. Going back to time domain, we get:
$$\epsilon''(t -t') = \frac{\sigma}{\epsilon_0} \, \theta(t - t')$$
which gives the following relation between the polarization and the electric field:
$$\mathbf{P}(t) = \sigma \, \int_{-\infty}^{t}{\mathbf{E}(t') \, dt'}$$
or, the current density due to bound charges is:
$$\mathbf{J}(t) = \dot{\mathbf{P}}(t) = \sigma \, \mathbf{E}(t)$$
But, this is just Ohm's law in differential form, provided the "bound" charges are the free ones, as is the case for conductors.