💼 Case: lets consider two positively charged particles with initial condition $\vec P_1$ moving towards $\vec P_2$ which is stationary.
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
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
$$ \partial _\mu j^\mu =0 $$
💼 Case: consider an infinitesimal volume $\delta \tau '$ that is displaces by $\delta \vec r'$ in a field $\vec E$
The charge of the volume is
$$ q=\rho \delta \tau' $$
The rate of change of the work done by the $\vec E$ field is
$$ \delta W=\rho \delta \tau'\text d \vec r'\cdot \vec E $$
If we integrate with respect to time we get
$$ \vec E\cdot \vec v \rho=\vec E \cdot \vec j = \vec E \cdot \left [\frac{1}{\mu_0} (\vec \nabla \times \vec B)-\epsilon_0 \left ( \frac{\partial \vec E}{\partial t} \right ) \right ] $$
where we used one of maxwells equations
🤛 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