How Does Poynting's Theorem Explain Ohmic Loss?

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The discussion centers on Poynting's Theorem and its interpretation in the context of electromagnetic fields. The equation presented highlights the relationship between the electric field (E) and current density (J), indicating that the term E·J represents ohmic loss, as noted in Cheng's "Field and Wave Electromagnetics." There is a query regarding the absence of this interpretation in Griffiths' work and a request for clarification on the meaning of the J·E term. Participants seek to understand whether the focus is on the interpretation of the term itself or the derivation of its connection to ohmic loss. Overall, the conversation aims to clarify the significance of the J·E term in the context of energy loss in electromagnetic systems.
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Poynting Theorem:

\frac {dW}{dt} \;=\; \int _{v'} (\vec E \cdot \vec J) d \vec {v'} \;=\; -\frac 1 2 \frac {\partial}{\partial t} \int _{v'} ( \epsilon_0 E^2 +\frac 1 {\mu_0} B^2) d \vec {v'}\;-\;\frac 1 {\mu_0} \int _{s'} (\vec E X \vec B) d \vec {s'}

In Cheng's "Field and Wave Electromagnetics", it interpret this is ohmic loss because:

\vec E \cdot \vec J \;=\; \sigma E^2

Which is the ohmic loss.

I don't see it described like this in Griffiths. Can anyone comment what this term really means?
 
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What is your question, is it how to interpret the J.E-term (. denotes scalar product), or how you find out that the J.E-term is the Ohmic loss?
 
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