You would be known about exponent of power on the dimension, that gives w^1.

However, I'm not sure that such calculations will always be correct, because it is necessary to consider the anisotropy of the medium, skin effect and the comparison of frequency with the plasma frequency. To determine the contribution of dielectric loss and conductivity (with  account the skin effect) from pure microwave measurement would be very difficult.

Plese have a look at the following discussion:

https://www.researchgate.net/post/Dielectric_Constant_F_m_vs_Conductivity_S_cm

If you have done Impedance Spectroscopy, you probably calculated the resistance and capacitance of the sample (using the Cp-Rp mode of the analyzer), which allows you to determine its ε' and ε''. 

Therefore, you can calculate the ac conductivity as following:

σac = ε''.ω.ε0

which is the same as using: σac = ε׳.tan(δ).ω.ε0

since tan(δ) = ε''/ ε'

You can take a look at my articles in each we study dielectric materials using impedance spectroscopy.

I m providing the link of two works where same thing (σac)  is measured with two different equation. 

http://www.sciencedirect.com/science/article/pii/S030488531200830X

http://www.sciencedirect.com/science/article/pii/S0925838815313025

I believe that expressions (6) and (7) (in the article Structural, morphological, electrical and magnetic properties of Dy doped Ni–Co substitutional spinel ferrite, http://www.sciencedirect.com/science/article/pii/S030488531200830X) contradict each other, and equation (7) [relation (6) in the cited paper [ref.21], Structural and electrical properties of nickel cadmium ferrites prepared through self-propagating auto combustion method] also contains an error. Moreover, the link to article [22] (The effect of Mn substitution on the magnetic and dielectric properties of cobalt ferrite synthesized by an autocombustion route) says: “shows the variation of log σac with log ω2, which are linear, indicating that the conduction is due to small polarons [22]”. However, this statement contradicts the statement about the linear dependence of σ(ω) contained in [22]: "The linear variation of AC conductivity with frequency confirms small polaron type of conduction mechanism".

Note also that the reviewed articles use different designations of the same values. All expression are given in the SI system, where eps0 (ϵ0 or ε0) is the electromagnetic constant (permittivity of vacuum / air / free space). Usually full (complex) permittivity (dielectric constant) is defined as ε’ – i*ε" where the loss is primarily responsible imaginary part (ε”), i=(-1)1/2, the tangent of dielectric loss tan δ = ε"/ε' or e”/e’. In mentioned articles the real part of permittivity ε' is marked as ε or εr (relative dielectric constant).

See also discussion:

https://www.researchgate.net/post/Why_imaginary_part_of_dielectric_constant_Vs_frequency_show_a_decreasing_behavior_in_low_frequency_region

Then what will be the exact equation for the ac conductivity ?

I believe that in some conditions other models can work well also. In particular, this can be the models that include additional terms with a quadratic dependence on frequency in a certain frequency range. However, in this case, the coefficient in front of ω2 should be different from εϵωtanδ.

So, various mechanisms can work in different ranges depending on the material composition and structure [see, for example, our papers:

https://www.researchgate.net/publication/290055764_Dielectric_and_Magnetic_Properties_of_Two-Phase_Composite_System_Mn-Zn_or_Ni-Zn_ferrites_in_Dielectric_Matrices?ev=prf_pub;

https://www.researchgate.net/publication/268787728_Modeling_the_structure_and_conditions_of_the_absorption_of_electromagnetic_radiation_in_magnetic_composites_using_effective_medium_approximations?ev=prf_pub].

If grain boundaries of your ferrite material have a conductivity less than grains of ferrite, the ferrite can also be considered as a composite material and apply the various models of the effective medium.

Article Dielectric and Magnetic Properties of Two-Phase Composite Sy...

Article Modeling the structure and conditions of the absorption of e...

1st formula which you can mentioned in your questions can be used.

you can analyse dielectric loss as a function of frequency. from that you can calculate dielectric constant and then use it in the formula you have shown above.

σ = ωɛoɛtanδ , this formula is correct,please give reference

Dear Ashok KR Pradhan,

I am also thinking on this question.

If Z = R+ jX be the impedance of a Dielectric of thikness d and area A, the admittance Y is given by Y = G+jB.

G ( conductance) = R/(R²+X²) and susceptance B = - X/(R²+G²).

From impedance meter I have R and X and hence calculate conductivity, σ = G. d / A is used.

What's wrong with this?

Please discuss.

Thanks and Regards

N Das