Deriving the Poynting theorem in a linear medium

[MatinSAR]

Homework Statement

Derive related equation using maxwell equations and describe each term.

Relevant Equations

Poynting Theorem.

Hello, I have two questions. First, when deriving $\nabla \cdot \mathbf{S} = -\frac{\partial u}{\partial t} - \mathbf{J} \cdot \mathbf{E}$, should we consider a linear medium? When we subtract $\mathbf{E} \cdot (\nabla \times \mathbf{H})$ from $\mathbf{H} \cdot (\nabla \times \mathbf{E})$, we get $-\mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t} - \mathbf{E} \cdot \frac{\partial \mathbf{D}}{\partial t} - \mathbf{J} \cdot \mathbf{E}$. Knowing that $\frac{d}{dt} (\mathbf{H} \cdot \mathbf{B}) = \frac{d}{dt} \mathbf{H} \cdot \mathbf{B} + \mathbf{H} \cdot \frac{d}{dt} \mathbf{B}$, for a linear medium with $\mathbf B=\mathbf {\mu} \mathbf H$ we can rewrite $\mathbf{H} \cdot \frac{\partial \mathbf{B}}{\partial t}$ as $\frac{1}{2} \frac{d}{dt} (\mathbf{H} \cdot \mathbf{B})$ or $\frac{d}{dt} u_B$ because for a linear medium we can say that $\frac{d}{dt} \mathbf{H} \cdot \mathbf{B}=\mathbf{H} \cdot \frac{d}{dt} \mathbf{B}$.

Applying the same process for the $\mathbf E$ part, we can derive the $\frac{d}{dt} \mathbf{u_E}$ term. Then we write $\mathbf u=\mathbf{u_E} + \mathbf{u_B}$.
Is my understanding correct?

Second question: Is $\nabla \cdot \mathbf{S}$ the net energy flux passing a point? What does the term $- \mathbf{J} \cdot \mathbf{E}$ represent? I know that $\mathbf{J} \cdot \mathbf{E}$ is the work done by the field to move charges, but what does the negative sign indicate?

 

[anuttarasammyak]

Homework Statement: Derive related equation using maxwell equations and describe each term.
Relevant Equations: Poynting Theorem.

Second question: Is ∇⋅S the net energy flux passing a point? What does the term −J⋅E represent? I know that J⋅E is the work done by the field to move charges, but what does the negative sign indicate?

Divergence is generation of energy. The sum of the three equals zero means energy conservation.

 

[MatinSAR]

Divergence is generation of energy.

Like when we have an emf in the region?

 

[anuttarasammyak]

Yes, decrease of u generates S to go.

 

[MatinSAR]

Yes, decrease of u generates S to go.

Thank you for your help.

 

[anuttarasammyak]

Homework Statement: Derive related equation using maxwell equations and describe each term.
Relevant Equations: Poynting Theorem.

Hello, I have two questions. First, when deriving ∇⋅S=−∂u∂t−J⋅E, should we consider a linear medium?

Permeability is defined by ratio of B and H. It is not always constant but can be complex function of multi-variables as ferro magnetism shows. Here its zero time derivative ,i.e. constant as for time, is used.