# Are P and E fields in LIH dielectric in dynamic equilibrium

1. Nov 29, 2013

### sigma_

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

P = ε0χeE.....(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?

2. Nov 30, 2013

### Jano L.

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 $\mathbf D = \epsilon_0 \mathbf E_0$.

3. Nov 30, 2013

### sigma_

We also know, however, that

∫D.dA = Qfenc

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?

Last edited: Nov 30, 2013
4. Nov 30, 2013

### 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 $\epsilon$.

5. Dec 6, 2013

### sigma_

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?

6. Dec 6, 2013

### Jano L.

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

7. Dec 6, 2013

### sigma_

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

8. Dec 7, 2013

### 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.

9. Dec 7, 2013

### sigma_

Thank you very much for your help!