# Mechanical analogue of permittivity

1. Jan 12, 2014

### Saw

I refer to the velocity of propagation of an EM wave in a medium.

I have been wondering about what plays the role, in this context, of inertia and elasticity. Here the formula has nothing in the numerator and the denominator is the product of the electric permittivity by the magnetic permeability of the material in question.

Initially I thought that permittivity and permeability played the role of inertia, because they occupy the denominator (the higher they are, the lower the velocity) and also because some books speak of “optical density”.

But then I heard the contrary opinion and also realized the following: permittivity is the facility of the atoms (or atoms network if you wish) to form dipoles and polarization is a sort of elastic tension; so what happens with this concept is that we are talking about something akin to Young modulus (in a solid) or Bulk modulus (in a fluid), it is only that the latter take the perspective of difficulty of deformation whilst permittivity would be easiness of deformation.

This would make sense in terms of dimensions since elasticity modulus is Newtons x m-2, whilst permittivity is Newtons -1 x m-2.

However, the dimensions of permittivity also have charge squared. And in any case inertia must be somewhere. So can it be then that permittivity is an empirical concept that mixes the two components: elasticity (or rather its opposite) and a sort of inertia?

2. Jan 14, 2014

### Malverin

Permittivity and permeability are something like inertia yes.

There are even some toughts, that electromagnetism is the cause for inertia

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

electromagnetic self force

3. Jan 14, 2014

### Saw

Thank you. That is very interesting. I gather from reading the wiki entry that actually mass comes from EM energy, but not only, though in any case you need the Higgs to explain the mass of elementary particles...(?). But that goes beyond the scope of the original question.

But the question was whether they (ε and μ) are the two things: inertia, yes, but at the same time not elasticity, but the opposite.

By this I mean that:

- a medium is said to be elastic (and hence transmits waves very quickly) when it is hard to deform it but for the same reason it recovers the position of equilibrium very quickly, with "springiness";
- instead a medium with high ε or μ would get easily deformed (i.e. polarized) and would return slowly to its equilibrium position (absence of dipoles).

This is also true, isn´t it?

4. Jan 14, 2014

### sophiecentaur

If you get too fixed on making an analogue fit, you can end up with confusion because it's only an analogue, in that it shares the same Equations over some range and conditions. There are two formula for wave speed. One for EM and one for Mechanical waves, the constituent parts can be paired off with one another just by the resemblance in the form of the equations. I don't think there needs to be much more in it than that. A useful tool for thinking about a new problem in terms of a familiar one. Good value but nothing to lose sleep over.

Let's face it, you could write an equation for a wave that never exists in any physical form at all. The Maths would still work but what would the result signify?

5. Jan 14, 2014

### Saw

Very true. My favorite game is precisely checking to what point analogies can be stretched out without breaking down. But in this case I think it can be fruitful to explore further.

.. which is not little. Detecting the analogy and playing with it has a didactic value, it is like a sort of mnemonic trick.

But if there is familiarity and resemblance between two components, it is not just because they occupy the same place in the equations. That happens because the underlying physical mechanism is similar. Precisely what is funny with permittivity is that it does NOT occupy the same place in the equation as for example the force of tension, but (when conveniently unmasked) it turns out to be a close relative of tension. Please note also that, as the textbooks state, elasticity is ultimately an electrostatic phenomenon.

Actually, I was thinking that the title of the thread may be a little misleading, since really here there is not so much of an "analogy" but variations of the same phenomenon. You can say that the propagation of an EM wave in vacuum is a different world. But its transmission through a material medium is something involving vibrating matter and that, yes, through electromagnetic influence but that also happens at microscopic level with a string or sound wave.

Sure! But before going to sleep, I have realized that the answer may be the following: electric permittivity of a material (which could also be called “capacitivity” as tiny-tim once said) may play the role of “(non-)rigidity” and in turn magnetic permeability (which could also be called “inductivity”) would play the role of inertia. How do you see that?

6. Jan 15, 2014

### Malverin

Why not elasticity?
They resist change in energy.
It is the same with a spring for example.
If you stretch it or compress it it will resist and it will try to restore its equilibrium state.

7. Jan 15, 2014

### Saw

Yes, that is what happens with dipoles, for example. They do resist changes. But the question is, given this, what does permittivity measure: how much they resist or rather how much they yield, how hard it is is to apply a change on them or rather how easy it is?

I think that common language is a little tricky in this field. In day-to-day language we say that something is elastic when it streches out or compresses instead of breaking or simply remaining rigid. In physics, it seems to me that the concept means, assuming that the material does not break, "how rigid it is". Permittivity, instead, looks like something similar but regarded from the other side: assuming (again) that a material does not "break" (and conduct electricity), "how non-rigid it is".

8. Jan 15, 2014

### Malverin

The μ and ε are like spring constant => when they are greater it is harder to disturb the equilibrium state.

When they are greater, the speed of waves will be smaller. So the disturbance in equilibrium will travel less distance for given amount of time.

Last edited: Jan 15, 2014
9. Jan 15, 2014

### Saw

No, you have to choose one or the other: if permittivity is like the spring constant, or rather like Young modulus (because we talk here about a material, not a specific object), then when it is greater, the speed of the wave should also be bigger, not smaller.

10. Jan 16, 2014

### Malverin

Maybe I am wrong about elasticity analogy for permittivity, but for mechanical system
greater spring constant => slower deformation

for example

If we have a vibration source with power P = 1 W
and a spring with constant K = 1 N/m
And we use this source to deform this spring with 1 meter

The mean value of the force will be

Fm = (k*dL)/2 = 1*1/2 =1/2 N

an the mean speed of deformation =>

Vm = P/Fm = 2 m/s

And when we have a spring with greater constant. For example
K = 2 N/m

then

Fm = (k*dL)/2 = 2*1/2 =1 N

and mean speed of deformation will be

Vm = P/Fm = 1 m/s

11. Jan 16, 2014

### Saw

I am not sure about your calculations, but it is clear that all formulas about speed of waves have the corresponding elasticity force in the numerator, meaning that the higher such force is (higher resistance to deformation and hence recovery of equilibrium also with higher force), the *faster* that the velocity is. See for example:

String waves --> v = sqr(tension/linear density)
Sound --> v = sqr(compressibility/volume density)

In the case of a spring you woud rather talk about frequency or angular velocity (ω) and again here k (stifness constant) is in the numerator:

Spring --> ω = sqr(k/m)

In an LC oscillator, when defining its angular velocity, you would not use permittivity and permeability because the latter refer to a material and you need the equivalent for a specific object. Thus you get capacitance and inductance, the formula being:

ω = sqr(1/LC)

But you can also put it like this, as I have just seen in a book, by the way:

ω = sqr(1/C/L)

This would lead us to accept this suggestion I was making for the speed of an EM wave in a medium (a pure mnemonic rearrangement):

EM wave --> v = sqr(1/ε/μ)

12. Jan 16, 2014

### Malverin

Yes you are right.

1/(εμ) is something like stiffness.

May be if we can isolate electric or magnetic component and test different mediums we can get a better sense what they are.
I don't think we can isolate the magnetic only, because any changing magnetic field induces currents.
But maybe we can isolate the electric part.
If we use a charged sphere for example, and we apply unipolar high voltage pulses to it.
I think there will be something like acustic pressure waves that will affect dielectric and conductors, but there will be no magnetic field (no moving charges there. But there will be some moving charges in the medium, so I am not sure...)
And then we can use different mediums (with the same ε and we change the μ only and then the opposite). So we can determine what is their influence to electric/magnetic components.

Last edited: Jan 16, 2014
13. Jan 18, 2014

### Saw

That attempt (isolation of electric from magnetic effects) looks difficult, but measuring the distinct effects of permittivity and permeability is something you could do in a LC circuit, wouldn’t you?, since permeability increases inductive reactance (which delays current) whereas permittivity reduces capacitative reactance (which delays tension).

On another note, I have checked that the inverse of stiffness is a coined concept, called “compliance”. Thus could we talk about the “compliantivity” = “capacitivity” = permittivity of a given material?

14. Jan 18, 2014

### Malverin

Something like that.
It is interesting that when we have $\ \ \textit{}\mu\ >\ \mu_0$ , and/or $\ \ \textit{}\epsilon\ >\epsilon_0$ , magnetic interaction influence becomes stronger than electrostatic =>

magnetic force gets stronger with $\ \ \textit{}\mu$ rising, and electrostatic - gets weaker with $\ \ \textit{}\epsilon$ rising

15. Jan 19, 2014

### Saw

That is another interesting point about these two concepts... and there happens to be something I don't understand here.

The physical mechanism behind permittivity is clear to me. If the electrostatic field has to traverse a dielectric, the latter gets polarized (the more, the higher its permittivity)= dipoles form in it (if they did not exist before) and in any case they get aligned in the direction of but against the source (electric) field, thus weakening it.

And I read that a ferromagnetic material inside a magnet gets magnetized (the easier, the higher its permeability) = its domains get aligned with the source (magnetic) field... thus reinforcing it? But what about Lentz law? Shouldn't the induced magnetic field oppose the source field?

16. Jan 19, 2014

### Malverin

It looks odd on first sight, but in fact it is normal.
When magnetic domains align to external magnetic field, they amplify it.
And when electric dipoles align to external electric field, they weaken it.
So because of that, force becomes stronger with μ rising and weaker with ε rising.

And maybe electromagnetic waves travel slower with greater μ and ε ,
because they interact with more domains(dipoles) and that slows them more.

Last edited: Jan 19, 2014