# Is the effective permittivity of a conductor imaginary number or negative real?

I am interested in how dielectrics and conductors, and materials between them, should be treated in Maxwell's equations.

Let's consider Ampere's law in a material with finite conductivity $$\sigma$$ in frequency domain: $$\nabla\times{\bf H}=\sigma{\bf E} +i\omega\epsilon_0\epsilon_r{\bf E}\\ =i\omega\epsilon_0\left(\epsilon_r+\frac{\sigma}{i\omega\epsilon_0}\right){\bf E}\\ =i\omega\epsilon_0\epsilon_\text{eff}{\bf E}$$ $$\left(\epsilon_\text{eff}:=\epsilon_r+\frac{\sigma}{i\omega\epsilon_0}\right)$$

This indicates that a material with conductivity $$\sigma$$ and permittivity of $$\epsilon_r$$ can be can be interpreted as a material with $$0$$ conductivity and permittivity $$\epsilon_\text{eff}$$.

When $$\sigma$$ is large enough, $$\epsilon_\text{eff}$$ will be imaginary: $$\epsilon_\text{eff}\approx\frac{\sigma}{i\omega\epsilon_0}$$

Therefore, I thought it appropriate to consider a good conductor as a dielectric with a large imaginary part.

On the other hand, however, Drude's equation about the permittivity of metals is: $$\epsilon_\text{metal}=1-\left(\frac{\omega_p}{\omega}\right)^2$$ This is to say that (in the low frequency range) metals behave as if it has a negative real permittivity.

(Although not about metals, I have done electromagnetic simulations on plasmas with negative permittivity using Drude's equation, which explained the actual phenomena well.)

Obviously, the value of $$\epsilon_\text{eff}$$ is imaginary, but of $$\epsilon_\text{metal}$$ is a negative real number. These relationships perplex me because I thought that taking the limit of increasing $$\sigma$$ in the $$\epsilon_\text{eff}$$ formula would naturally be consistent with the metal case.

I think there is something fundamentally wrong with me, but what mistake am I making?

Postscript:

I am trying to solve and simulate Maxwell's equations in the microwave band. The system to be solved contains a conductor with halfway conductivity.

When modeling this system, I am confused as to whether the effect of conductivity should be included in the real or imaginary part of the dielectric constant.

Since I have an electromagnetic simulator(HFSS), I can set the conductivity and dielectric constant arbitrarily and get some solution, but I need to understand if those settings are appropriate.

I made a simple model of a coaxial waveguide on the simulator and placed a dielectric with permittivity $$\epsilon_r$$ in place of the inner conductor. I ran both settings, $$\epsilon_r=-100$$ and $$\epsilon_r=1-100i$$, and both showed coaxial waveguide-like propagation, similar to the metal case. (Actually, I set $$\tan\delta=-\epsilon_r''/\epsilon_r'=100$$ because the simulator cannot handle complex parameters.) This result confirms my expectation that either would not work as a waveguide, but the wave seems to be attenuated when set as $$\tan\delta$$.

• Your derivation combining $\sigma$ and $\epsilon_r$ ignores that these quantities depend on frequency. The Drude model does not, in fact it derives how they should (in the simplest model) depend on frequency. Nov 14, 2023 at 5:42
• I know that for a very wide range, parameters such as permittivity exhibit a very complex frequency response. What I want to know is how to interpret the conductivity when the frequency is limited (e.g., microwave band). I have added a postscript to my question. Nov 14, 2023 at 6:55
• > "I am confused as to whether the effect of conductivity should be included in the real or imaginary part of the dielectric constant." Both - AC current, in general, is not in phase with electric field, so it can be described by complex conductivity $\sigma = \sigma'+i\sigma''$ with non-zero imaginary component $\sigma''$. Nov 14, 2023 at 16:21

## 1 Answer

Usually, the Drude model amounts to adding a relaxation time $$\tau$$ corresponding to a linear damping force of the charge carriers.

Your first approach corresponds to the low frequency limit $$\omega\tau\ll1$$. In this regime, there is finite real conductivity $$\sigma_0$$ which corresponds imaginary, divergent permittivity.

Your second approach essentially captures plasma oscillations. Technically, it only depends on the restoring electrostatic force. It does not require the damping Drude force, and corresponds to the high frequency regime $$\omega\tau\gg1$$.

You can interpolate the two by solving the linearised, Drude's model: $$\sigma = \frac{\sigma_0}{1+i\omega \tau}$$ This gives the effective relative permittivity: $$\epsilon_e = \epsilon_r\left(1+\frac{\sigma_0/\epsilon}{i\omega(1+i\omega\tau)}\right)$$ As you can see, your permittivity formula for metals is obtained precisely when $$\omega\tau\gg1$$ and identifying the plasma frequency: $$\omega_p = \sqrt{\frac{\sigma_0}{\epsilon\tau}}$$

Thus, it is normal that the two regimes to not agree. They are precisely at opposite ends of the spectrum with respect to $$\tau$$. Depending on your frequency band, you should use one or the other, but not both at the same time.

Hope this helps.