Understanding the Concept:
According to Maxwell's equations, the total current density in a lossy dielectric medium consists of two main components:
• Conduction Current Density (\( J_c \)): Arises due to the motion of free charges under an electric field, defined by Ohm's Law in point form as:
\[
J_c = \sigma E
\]
• Displacement Current Density (\( J_d \)): Arises from time-varying electric displacement fields, given by:
\[
J_d = \frac{\partial D}{\partial t} = \varepsilon \frac{\partial E}{\partial t}
\]
Step 1: Compute the expressions for both current densities
Given the time-varying electric field expression:
\[
E = E_0 \cos(\omega t)
\]
Substitute \( E \) into the expression for conduction current density:
\[
J_c = \sigma E_0 \cos(\omega t)
\]
The maximum peak amplitude of the conduction current density is:
\[
|J_c|_{\max} = \sigma E_0
\]
Next, substitute \( E \) into the expression for displacement current density:
\[
J_d = \varepsilon \frac{\partial}{\partial t} [E_0 \cos(\omega t)] = \varepsilon E_0 (-\omega \sin(\omega t)) = -\omega \varepsilon E_0 \sin(\omega t)
\]
The maximum peak amplitude of the displacement current density is:
\[
|J_d|_{\max} = \omega \varepsilon E_0
\]
Step 2: Determine the ratio of the amplitudes
Now, evaluate the ratio between the maximum conduction amplitude and the displacement amplitude:
\[
\text{Ratio} = \frac{|J_c|_{\max}}{|J_d|_{\max}} = \frac{\sigma E_0}{\omega \varepsilon E_0} = \frac{\sigma}{\omega \varepsilon}
\]
This algebraic ratio represents the loss tangent (\( \tan\delta \)) of a material medium. The result matches Option (B).