Talk:Poynting's theorem

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$\rho =\varepsilon _{0}{\vec {\nabla }}\cdot {\vec {E}}$ Gauss law

${\begin{aligned}\mu _{0}{\vec {J}}\cdot {\vec {E}}&=\left({\vec {\nabla }}\times {\vec {B}}-\mu _{0}\varepsilon _{0}{\frac {\partial {\vec {E}}}{\partial t}}\right)\cdot {\vec {E}}\\&=-\nabla \cdot ({\vec {E}}\times {\vec {B}})+{\vec {B}}\cdot \left({\vec {\nabla }}\times {\vec {E}}\right)-\mu _{0}\varepsilon _{0}\left({\frac {\partial {\vec {E}}}{\partial t}}\cdot {\vec {E}}\right)\end{aligned}}$ aligned version

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

references

Wikiversity has a number of good articles that might help you understand Poynting's theorem. But I don't like Wikipedia:Poynting's theorem.

Without stress tensor

• https://ocw.mit.edu/resources/res-6-001-electromagnetic-fields-and-energy-spring-2008/chapter-11/11.pdf
• Wikipedia:Poynting's theorem
• Wikipedia:Poynting vector

With stress tensor

• https://ocw.mit.edu/courses/nuclear-engineering/22-105-electromagnetic-interactions-fall-2005/readings/chap3.pdf
• Wikipedia:Stress–energy tensor

trash

$\mu _{0}{\vec {J}}\cdot {\vec {E}}=\left({\vec {\nabla }}\times {\vec {B}}-\mu _{0}\varepsilon _{0}{\frac {\partial {\vec {E}}}{\partial t}}\right)\cdot {\vec {E}}$

$\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}$