💼 Case: lets consider two positively charged particles with initial condition $\vec P_1$ moving towards $\vec P_2$ which is stationary.

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Global conservation

Classically this is action at a distance which allows the transfer of momentum from $m_1$ to $m_2$ as long as $\Delta \vec P_i=-\Delta \vec P_2$ at each instant in time. Ie conservation of momentum

This is an example of Global conservation

🗒️ Note: global conservation is not possible in relativity

Local conservation

In special relativity only events that occur at the same point in space can unambiguously be called simultaneous.

Local conservation law is a law that is conserved at a point in spacetime

🗒️ Note: local conservation is a stronger requirement than global conservation

💃 Example: Conservation of charge previously seen is a local conservation law

$$ \partial _\mu j^\mu =0 $$

Local conservation of energy in electrodynamics

💼 Case: consider an infinitesimal volume $\delta \tau '$ that is displaces by $\delta \vec r'$ in a field $\vec E$

🤛 Lets look at the right hand side

$$ -\epsilon_0 \vec E \cdot \frac{\partial \vec E}{\partial t} =-\frac{\epsilon_0}{2} \frac{\partial}{\partial t} (E^2) $$

🤜 Now the left hand side

$$ \begin{aligned} \vec E \cdot (\vec \nabla \times \vec B )&=- \vec \nabla \cdot (\vec E \times \vec B)+ \vec B\cdot \left ( -\frac{\partial \vec B}{\partial t}\right ) \\ &=-\vec \nabla \cdot (\vec E \times \vec B)-\frac 12 \frac{\partial}{\partial t} (B^2) \\

\end{aligned} $$

🤲 Putting it together we get

$$ \begin{aligned} \vec E \cdot \vec j = - \vec \nabla \cdot \overbrace{\left ( \frac{1}{\mu_0}\vec E \times \vec B \right )}^{\vec S}- \frac{\partial}{\partial t} \overbrace{\left ( \frac{\epsilon_0}{2} E^2 + \frac{1}{2\mu_0} B^2 \right )}^{u} \\ \boxed{\vec E \cdot \vec j+ \vec \nabla \cdot \vec S+ \frac{\partial u}{\partial t} =0}\qquad \qquad \quad \end{aligned} $$

where $\vec S$ is the energy flux density (Poynting vector) and $u$ is the energy density in the fields