MatinSAR
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- 204
- 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?
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?
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