A discrepancy on Wikipedia or a misunderstanding on my part?

Question

Recently I have been trying to understand complex magnetic impedance and when I try to formulate the equations for it based off the similar equations for electrical circuits I keep coming up with some weird answers that don't make sense. I realize that two of the equations regarding electrical fields I found on Wikipedia seem logically inconsistent and this may be the root of the problem as I depend on them in my formulations.

On this page on Wikipedia two related equations are defined relative to the tangent loss angle:

https://en.wikipedia.org/wiki/Dielectric_loss#Electromagnetic_field_perspective

The two equations are:

$$\tan(\delta) = \frac{\omega\epsilon'' + \sigma}{\omega\epsilon'}$$

$$tan(\delta) = \frac{\epsilon''}{\epsilon'}$$

However I notice if I set these equations equal to each other and simplify I get the following.

$$\frac{\omega\epsilon'' + \sigma}{\omega\epsilon'} = \frac{\epsilon''}{\epsilon'}$$

$$(\omega\epsilon'' + \sigma) \epsilon' = (\omega\epsilon')\epsilon''$$

$$\omega\epsilon'\epsilon'' + \sigma\epsilon' = \omega\epsilon'\epsilon''$$

$$\sigma\epsilon' = \omega\epsilon'\epsilon'' - \omega\epsilon'\epsilon''$$

$$\sigma\epsilon' = 0$$

But this can't possibly be correct can it? This would imply that either $\epsilon'$ or $\sigma$ must always be 0. However if $\epsilon'$ is ever 0 then both equations becomes undefined. Also if $\sigma$ is always 0 then the material would have to be a perfect resistor for the equations to make sense and also why even including conductivity as a variable at all in that case?

I am convinced I must be doing something wrong but I have no idea what that could be.

Answer 1

Initial answer:

The second equation is an approximation used when conductivity ($\sigma$) is very small.

You can see that in the limit $\sigma\to 0$ the two equations become equivalent.

The approximate form is appropriate in vacuum, air, and many other good insulators.

Alternate answer:

After reading the Wiki page, it appears the Wiki authors actually had a different interpretation in mind.

In the discussion of the first equation, they first mention

The permittivity can have real and imaginary components (the latter excluding ''σ'' effects, see below)

And then define

ε′′ is the imaginary component of permittivity attributed to bound charge and dipole relaxation phenomena [i.e. conductivity effects, caused by free charge, are excluded],

But also comment that the losses due to the bound charge are "indistinguishable from the loss due to the ''free'' charge conduction that is quantified by σ."

In the discussion of the second equation they refer to another article where they define $\varepsilon''$ differently, according to

$$\varepsilon_r = \varepsilon_r' - \frac{i\sigma}{\omega\varepsilon_0} $$

That is, in this version they simply include the conductivity effects in $\varepsilon''$ rather than accounting for them separately.

So really, it seems the intent of the Wiki authors is to define $\varepsilon''$ differently in the two equations, in one case excluding free charge effects and in the other including them.

The equations are equivalent, if you realize that certain symbols are used with different meanings in the two cases.

But then, there is a further comment in the Wiki article,

In other words when the dielectric is a good insulator, and thus ''σ'' is very small, then the above equation two equations are a good approximation.

indicating that at least one Wiki contributor had my initial explanation in mind to explain the relationship between the two equations.