# About polarization and electric field

1. Sep 26, 2009

### KFC

In the text, it is said that the polarization is just the response of the input electric field so they have

$$P = \epsilon_0 \chi E$$
where P is the polarization and E is the input.

This makes sense to me. However, why we need $$\epsilon_0$$ sitting there? Since $$\epsilon_0$$ has no dimension, so even there is no $$\epsilon_0$$, the relation is still the relation b/w input and response. So what's the different if we have $$\epsilon_0$$ there?

And by the way, the book said above equation is given when considering linear case. So what is the relation b/w polarization and field when there exists nonlinearity?

Thanks.

2. Sep 26, 2009

### kanato

$$\epsilon_0$$ has dimensions if you're not working in CGS units. It's a commonly used convention to write $$P = \epsilon E$$ and then replace $$\epsilon$$ with the value for the material at a given point in space, or $$\epsilon_0$$ for the vacuum, so $$\chi$$ is the ratio of the permeability of the material to the permeability of space in any unit system.

There's nothing you can really say about the nonlinear case in general. There, $$P = f(E)$$ where f is some function of the electric field. You could write it in the same form as above, like $$P = \epsilon_0 \chi(E) E$$ but this is done without loss of generality, so there is nothing learned. In real materials, what usually happens is that there is some electric field where the polarization saturates, and for E greater than that field, the polarization will be basically constant. And if you go much higher than that, then you will eventually get breakdown of the material and it begins to conduct. The same thing happens in the magnetic case, except for the breakdown. The linearity assumption is good up to moderately large fields in most cases.

3. Sep 26, 2009

### tiny-tim

Hi KFC!

(have an epsilon: ε and a chi: χ )

In SI units, the permittivity ε0 is measured in units of farad per metre.

The susceptibility χ is dimensionless.

Second-order susceptibility is a tensor, used in non-linear optics, with Pi = ε0χijkEjEk

Last edited: Sep 26, 2009
4. Sep 26, 2009

### KFC

Thanks. I still have a question. I learn from a text that the so-called absorption coefficient is related to the imaginary part of $$\chi(\omgea)$$, but in other books, they say the absorption coefficient should be

$$\frac{\omega}{c_0}\Im(\sqrt{1 + \chi_r + i\chi_{i}})$$

where $$\chi = \chi_r + i\chi_i$$, $$c_0$$ is the light speed in free space.

so which one is correct?

Last edited: Sep 26, 2009