Talk:Poynting's theorem

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$$\rho=\varepsilon_0\vec\nabla\cdot\vec E$$ Gauss law

$$\begin{align} \mu_0\vec J\cdot\vec E&=\left( \vec\nabla \times\vec B - \mu_0\varepsilon_0 \frac{\partial\vec E}{\partial t}\right)\cdot\vec E\\ &= -\nabla\cdot(\vec E\times\vec B)+\vec B\cdot\left(\vec\nabla\times\vec E\right)- \mu_0\varepsilon_0 \left(\frac{\partial\vec E}{\partial t}\cdot \vec E\right) \end{align}$$ aligned version

$$\vec f = \rho\vec E+\vec J\times\vec B$$ Force law

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$$\mu_0\vec J\cdot\vec E=\left( \vec\nabla \times\vec B - \mu_0\varepsilon_0 \frac{\partial\vec E}{\partial t}\right)\cdot\vec E$$

$$\mu_0\underbrace{\vec J\cdot\vec E}_{\partial u / \partial t} = $$$$ -\nabla\cdot\underbrace{\left(\vec E\times\vec B\right)}_{\mu_0\vec S} +\vec B\cdot\underbrace{\left(\vec\nabla\times\vec E\right)}_{-\partial \vec B / \partial t} -\mu_0\varepsilon_0 \underbrace{\left(\frac{\partial\vec E}{\partial t}\cdot \vec E\right)} _{\frac 1 2 \partial E^2/\partial t}$$