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Permittivity = EM conductivity?

  1. Mar 19, 2012 #1

    Saw

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    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 travelling 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”.
     
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  3. Mar 19, 2012 #2

    DrDu

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    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.
     
  4. Mar 19, 2012 #3

    tiny-tim

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    Hi Saw! :smile:
    Yes … R C and L have essentially similar effects in a circuit (if we regard reactance as similar to resistance :wink:),

    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! :biggrin:

    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). :smile:
     
  5. Mar 19, 2012 #4

    Saw

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    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...
     
  6. Mar 19, 2012 #5
    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?
     
  7. Mar 19, 2012 #6
    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.
     
  8. Mar 19, 2012 #7

    DrDu

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    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.
     
  9. Mar 19, 2012 #8
    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?
     
  10. Mar 19, 2012 #9

    tiny-tim

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    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! :wink:
    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 :wink:
    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
     
  11. Mar 19, 2012 #10

    DrDu

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    Which both factors?
     
  12. Mar 19, 2012 #11
    I meant permittivity and conductivity.
     
  13. Mar 19, 2012 #12

    DrDu

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    Then, as they are mathematically equivalent, there is only one factor.
     
  14. Mar 19, 2012 #13

    Saw

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    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:

    kmarinas86 pointed in the same direction here:

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

    However, DrDu seems later to identify both things:

    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 travelling 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?
     
    Last edited: Mar 19, 2012
  15. Mar 19, 2012 #14

    tiny-tim

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    are you misreading that as "permittivity and capacitivity"? :confused:
    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! :wink:

    (capacitance is the ability to separate charge … the capacitance of a material tells us how far these "bound" pairs are separated, per volt)
    i'll reply more, later
     
  16. Mar 19, 2012 #15

    DrDu

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    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.
     
  17. Mar 19, 2012 #16
    If you write down the fourth Maxwell's equation for a material with a (relative) permitivity tensor [itex]\hat{\epsilon}(\mathbf{x}, \omega)[/itex], and a conductivity tensor [itex]\hat{\sigma}(\mathbf{x}, \omega)[/itex] in frequency domain, you have:

    [tex]
    \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)
    [/tex]

    It is easy to see that you can atribute the conductivity tensor as an imaginary part of a complex permitivity tensor:
    [tex]
    \hat{\tilde{\epsilon}} \equiv \hat{\epsilon} + i \frac{\hat{\sigma}}{\epsilon_0 \, \omega}
    [/tex]
    or, attribute the permitivity tensor as an imaginary part of a complex conductivity tensor:
    [tex]
    \hat{\tilde{\sigma}} \equiv \hat{\sigma} - i \, \omega \, \epsilon_0 \, \hat{\epsilon}
    [/tex]
    These are two complementary ways to combine the same physics into one unified form ([itex]\hat{\tilde{\sigma}} = -i \, \omega \, \epsilon_0 \, \hat{\tilde{\epsilon}}[/itex]). 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.
     
  18. Mar 19, 2012 #17

    DrDu

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    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.
     
  19. Mar 19, 2012 #18

    Saw

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    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.
     
  20. Mar 19, 2012 #19

    tiny-tim

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    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 :wink:
     
  21. Mar 19, 2012 #20

    tiny-tim

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