How Do Conductivity and Permittivity Relate in Electromagnetic Theory?

[andyJ]

Hi,

I am trying to drive this famous equation: ε_r(ω) = 1 + iσ(ω)/(ε_0*ω)

First the regular solution:
J=∂P/∂t→In Fourier domain→J=iωP(ω)=iω{D(ω)-ε_0*E(ω)}; then replace J with σ(ω)*E(ω) and rearrange to reach the result

What makes me puzzle is we know that J_external is equal to σ(ω)*E(ω) but in foregoing solution I used J_internal which is is equal to ∂P/∂t

Here are definitions for all types of currents:

• J_total = -ε_0*∂E/∂t
• J_external = -∂D/∂t
• J_internal = ∂P/∂t

and
J_total = J_external + J_internal.

If the above definitions are correct then I can not drive ε_r(ω) = 1 + iσ(ω)/(ε_0*ω)

Any idea?

 

[eljose79]

Hello,

Thank you for your post. It seems like you are on the right track with your approach to driving the equation ε_r(ω) = 1 + iσ(ω)/(ε_0*ω). However, I think there may be a misunderstanding with the definitions of the different types of currents.

First, let's clarify the meaning of J_total, J_external, and J_internal. J_total represents the total current within a material, which is the sum of both the external and internal currents. J_external refers to the current that is generated by external sources, such as an applied electric field. On the other hand, J_internal represents the current that is generated by the material itself, due to the motion of charges within the material.

Now, let's look at the equation J=∂P/∂t. This equation is known as the continuity equation, which relates the current density (J) to the rate of change of polarization (P) in a material. This equation is valid for both the internal and external currents, as they both contribute to the total current density.

When you take this equation into the Fourier domain, you get J=iωP(ω). This equation is still valid for both the internal and external currents, as it is simply a transformation to a different domain.

Next, you replace J with σ(ω)*E(ω) and rearrange to reach the result ε_r(ω) = 1 + iσ(ω)/(ε_0*ω). This step is also valid, as you are simply substituting the current density with the appropriate expression for the current (σ(ω)*E(ω)).

Therefore, your approach is correct and you should be able to drive the equation ε_r(ω) = 1 + iσ(ω)/(ε_0*ω). I hope this helps to clarify any confusion. Keep up the good work!