# Can you calculate optical density from permittivity and permeability?

• Andrew Wright
In summary, when a substance is subject to a variable external field, its polarization, for one, does not obey the same proportionality factors as applied to constant field. For another, just adjusting the real permittivity values does not suffice, because the polarization turns out to depend on the past history of external field (though fortunately not of its future...).

#### Andrew Wright

Does electric perm and magnetic perm give you enough info to work out OD?

Do you think I have got the right forum for this topic?

The assumption i was making here is that the perms and OD relate to the speed of light in a substance.

I'm not sure to which physical situation you are referring to. I guess you mean that you neglect the imaginary part of the permittivity and then you have [EDIT: Corrected obvious typo in the final equation]
$$c=\frac{1}{\sqrt{\epsilon \mu}}=\frac{1}{\sqrt{\epsilon_0 \mu_0 \epsilon_{r} \mu_{r}}}=c_0/n \; \Rightarrow \; n=\sqrt{\epsilon_r \mu_r}.$$

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etotheipi
Cool :) Thanks.

Note the correction of the obvious typo! It's, of course,
$$n=\sqrt{\epsilon_r \mu_r}.$$

Is refractive index the same as optical density?

vanhees71 said:
As far as I know "optical density" is an oldfashioned expression for the absorption coefficient, i.e., it's rather related to the imaginary part of the refractive index, the extinction coefficient

And in fact the designation survives in nomenclature for "neutral density" attenuating optical filters.

Well, in some sense of course all this is related. One should be aware that this is about Green's functions in linear-response approximation, describing the propagation of electromagnetic waves in a medium, and the real and imaginary parts of the corresponding refraction index ##n(\omega)## as a function of frequency (neglecting spatial dispersion) are related by Kramers-Kronig relations, which hold very generally only based on causality.

hutchphd
Andrew Wright said:
Summary:: Hi. Is there a formula for getting OD from electric permativity and magnetic permeability?

Does electric perm and magnetic perm give you enough info to work out OD?

If I understand you correctly, 'yes'- most simply, the permittivity ε and permeability μ must be complex-valued. Optical density is associated with absorption (or scattering)- a scattering medium is heterogeneous, so in that case the refractive index may be real-valued but there is not a single-valued ε or μ for the entire sample.

hutchphd and vanhees71
...there is not a single-valued ε or μ for the entire sample.

Does this mean the perms have different values at different wavelengths?

Yes!

So, I've been thinking about this and permativity/permeability are properties that relate to electric and magnetic fields not historically about waves, which begs the question "wavelength of what exactly?" In the context of classical physics?

I'll keep trying. I know light is an em wave but the concepts of perms predate the understanding of em waves. So how can the values vary with wavelengths if really it is about fields that don't need wavelengths?

Light is an electromagnetic wave and to describe its propagation in the medium you can use Fourier transforms in terms of time. Then you have to solve the Maxwell equations in the medium for fields with harmonic time dependence only and then can build the waves in the time domain by the Fourier integral.

In the usual approximation you consider visible light and usual matter. The typical extension of an atom is much smaller than the wave length and thus you can treat the matter as homogeneous and that's why the permittivity and permeability can be considered as frequency dependent only (in the frequency domain). For a nice treatment in terms of classical physics, see e.g., the Feynman Lectures. The phenomenology is amazingly pretty accurate though of course the "true theory" is quantum mechanics:

https://www.feynmanlectures.caltech.edu/II_32.html

Andrew Wright said:
Does this mean the perms have different values at different wavelengths?

They can- that's called 'dispersion'. But different materials have different ε(λ) and μ(λ) as well. Discontinuities at material boundaries leads to scattering, which is not the same as absorption. OD is often used as a way to characterize turbid media.

Andrew Wright said:
I'll keep trying. I know light is an em wave but the concepts of perms predate the understanding of em waves. So how can the values vary with wavelengths if really it is about fields that don't need wavelengths?
When a substance is subject to static electric and magnetic fields, its polarization is proportional to the present (constant!) applied field.
But when the substance is subject to a variable external field, its polarization, for one, does not obey the same proportionality factors as applied to constant field. For another, just adjusting the real permittivity values does not suffice, because the polarization turns out to depend on the past history of external field (though fortunately not of its future...), even for the same values of external field.

Compare a current circuit - the mathematics of electromagnetic waves is actually somewhat analogical to current circuits!
In direct current circuits, resistors have a fixed and positive resistance, and current always flows in the direction of voltage.
In alternsting current circuits, however, resistance is no longer constant, or even required to be positive: an inductive coil can support current against voltage for some time after voltage reverses.

Many simple alternating current circuits can be approached as having complex resistance, though. (Many cannot!)

vanhees71
That's fascinating. Thank you :)