Are P and E fields in LIH dielectric in dynamic equilibrium

[sigma_]

If we know that the Polarization P in LIH dielectrics is proportional to the net field inside the dielectric according to:

P = ε₀χ_eE...(1)

And we know that

D = εE...(2)

Does it not follow that we can ascertain the polarization directly from the applied (free charge) field, since we can relate D to E, and then E to P?

The author of my electrodynamics text (Griffiths), says that we cannot. His explanation being that once we place dielectric in an external field E₀, the material will polarize and create an opposing field to the applied field, which in turn modifies the polarization again because it changes the total field that the atoms/molecules in the material are being subject to, and this process repeats over and over. In actuality, are these two quantities (E and P) in some sort of dynamic equilibrium within the material? If so, how come (1) and (2) are valid?

 

[Jano L.]

...In actuality, are these two quantities (E and P) in some sort of dynamic equilibrium within the material?...

Yes, only the equilibrium is more appropriately called static ($\mathbf E$,$\mathbf P$ are do not change in time).

If so, how come (1) and (2) are valid?

They both state the same thing: the polarization $\mathbf P$ is proportional to $\mathbf E$.

Does it not follow that we can ascertain the polarization directly from the applied (free charge) field, since we can relate D to E, and then E to P?

You seem to think that $\mathbf D$ is somehow simply related to applied field $\mathbf E_0$. In general, it is not! $\mathbf D$ is defined by

$$
\mathbf D = \epsilon_0 \mathbf E + \mathbf P,
$$

which contains both $\mathbf E, \mathbf P$, the "quantities in mutual equilibrium". There is no simple relation of $\mathbf D$ to $\mathbf E_0$, except inside a parallel plate capacitor, where $\mathbf D = \epsilon_0 \mathbf E_0$.

 

[sigma_]

Yes, only the equilibrium is more appropriately called static ($\mathbf E$,$\mathbf P$ are do not change in time).
They both state the same thing: the polarization $\mathbf P$ is proportional to $\mathbf E$.
You seem to think that $\mathbf D$ is somehow simply related to applied field $\mathbf E_0$. In general, it is not! $\mathbf D$ is defined by

$$
\mathbf D = \epsilon_0 \mathbf E + \mathbf P,
$$

which contains both $\mathbf E, \mathbf P$, the "quantities in mutual equilibrium". There is no simple relation of $\mathbf D$ to $\mathbf E_0$, except inside a parallel plate capacitor, where $\mathbf D = \epsilon_0 \mathbf E_0$.

Thanks for the reply, Jano!

We also know, however, that

∫D.dA = Q_fenc

so D has to somehow be related to the applied field, because it is the flux density of the free charge, and typically free charge is what we control, and create (apply) fields with.

I think maybe my interpretation of D is incorrect here then?

 

[Jano L.]

Both $\mathbf D$ and $\mathbf E_0$ due to free charges satisfy the same equation you wrote above, but that is not sufficient to establish simple relation between them. This is because it is just one equation and we deal with vectors, which have three components. The two quantities are different from each other by third vector field that has every surface integral zero. Such field is called solenoidal field, and can be very complicated. It will be determined by the arrangement of the free charges with density $\rho_{free}$ and the dielectric bodies. $\mathbf D$ at all points of the system can be found out from $\rho_{free}$, but knowledge of $\mathbf E_0$ won't help much; I think one would do better with some energy minimization method that takes into account the spatial dependence of $\epsilon$.

 

[sigma_]

Both $\mathbf D$ and $\mathbf E_0$ due to free charges satisfy the same equation you wrote above, but that is not sufficient to establish simple relation between them. This is because it is just one equation and we deal with vectors, which have three components. The two quantities are different from each other by third vector field that has every surface integral zero. Such field is called solenoidal field, and can be very complicated. It will be determined by the arrangement of the free charges with density $\rho_{free}$ and the dielectric bodies. $\mathbf D$ at all points of the system can be found out from $\rho_{free}$, but knowledge of $\mathbf E_0$ won't help much; I think one would do better with some energy minimization method that takes into account the spatial dependence of $\epsilon$.

So in the scenario I listed above, $\mathbf D$ and $\mathbf E_0$ are interchangable as long as a factor of $\epsilon$ is accounted for?

 

[Jano L.]

What do you mean by "interchangeable" ? They are not in simple relation one to another. What is your scenario concretely?

 

[sigma_]

What do you mean by "interchangeable" ? They are not in simple relation one to another. What is your scenario concretely?

Let's say, for instance, an LIH dielectric between the plates of a parallel plate capacitor.

 

[Jano L.]

In parallel plate capacitor, $\mathbf D = \epsilon_0 \mathbf E_0$ where $\mathbf E_0$ is the field due to free charges. The factor $\epsilon_0$ is permittivity of vacuum, not of the dielectric.

 

[sigma_]

In parallel plate capacitor, $\mathbf D = \epsilon_0 \mathbf E_0$ where $\mathbf E_0$ is the field due to free charges. The factor $\epsilon_0$ is permittivity of vacuum, not of the dielectric.

Thank you very much for your help!