# Relationship Between Conductivity and Lossiness of a material

I read that a material is loss-less if the conductivity is zero. I have always learned that conductivity is a measure of how easily the material can conduct a current. Does this then mean that the only materials that are loss-less are those that conduct no current? Can someone elaborate on this?

Edit: Can electromagnetic waves propagate through a loss-less media? How is this possible if there is 0 conductivity?

• A zero conductivity only implies no energy dissipation at DC. Even a non-conductive material will dissipate energy at high-enough frequency due to its nonzero polarization density P. Regarding your last point, EM waves even pass through vacuum in a completely lossless fashion. They don't even need a medium. Feb 4 '15 at 22:56
• Consider EM waves through water, which has a complex and frequency-dependent dielectric "constant", due to non-instantaneous orientational polarization. Sea water will also have conductivity, due to dissolved ions. Nov 9 '18 at 15:10

Ohmic dissipation is proportional to the square of the current multiplied by a resistance.

Yes, the resistance is inversely proportional to the conductivity, but the current induced by a given electric field is also proportional to the conductivity.

Hence the Ohmic losses end up being proportional to conductivity.

Symbolically:

$$P_{loss} \propto J^2 R$$ Where $J$ is the current density. But because $J = \sigma E$, where $\sigma$ is the conductivity and $E$ the electric field, and also $R \propto 1/\sigma$, then $$P_{loss} \propto \sigma E^2$$

Thus for a given electric field strength, materials with high conductivity dissipate more power. In terms of EM waves; a wave propagating in a conductor decays with an exponential scale length that decreases with increasing conductivity.

Electromagnetic waves propagate through lossless media because there is nothing to dissipate them! No currents are required for an EM wave to propagate; there are non in a vacuum...

I shall restrict this answer to only materials and their response to EM waves. After all conductivity is a material property. Furthermore for this handwaving analysis I shall stick to materials who have linear response to the fields (electric). The core argument still holds for non-linear responses but the analysis will differ.

To begin, imagine the material to be consisting of electrons that behave as springs if perturbed from their equilibrium position (linear response). So the electrons have some inherent frequency with which they vibrate around the nucleus. These oscillations are damped due to inter-electronic repulsion and other things. The external oscillating electric field then acts as a driving force to these electrons. This response of the electrons is what restricts the EM wave to propagate along the material. If the electrons did not interact with the field then the EM wave would propagate unhindered unrestricted to the material's geometry.

Thus there is inherent loss in the mechanism of conduction of EM waves in a material. In other words, conduction only occurs if there is a loss of energy that goes into oscillating the electrons that mediate the transfer.

Superconductors have infinite, rather than zero, conductivity at DC, but they are lossless for DC current.