Electrodynamics/Electric Energy

Introduction

Many objects in this universe are conservative, the charge of the universe for instance always remains constant. This is called Global Conservation of Charge. But also, if you take a defined area, then the charge in that also remains constant unless some is leaving through its boundary. And if some charge is moving through the boundary, then it must be equal to the change in charge in the volume. Just like if you and your friends are in a room, you cannot just not be in the room, you must go out the door.

This is called Local Conservation of Charge.

Charge Conservation

Let's look at the total charge inside a Volume V:

$Q(t) = \int_V \rho (\mathbf{r}) d \tau$

And going from what was said in the introduction and that the current travelling through a surface is:

$I = \int_S \mathbf{J} \cdot d \mathbf a$

We see that,

$\frac{dQ}{dt} = \int_S \mathbf{J} \cdot d \mathbf a$

Then by using the divergence theorem, we get the law of local conservation of charge

[Law of Local Conservation of Charge]

$\frac{d\rho}{dt} = - \nabla \cdot \mathbf{J}$

Conservation of Energy

[Law of Conservation of Energy]

$\frac{\partial}{\partial t} (u_{mechanical} + u_{electro magnetic}) = - \nabla \cdot \mathbf S$

Where

$\mathbf S = \frac{1}{\mu_0} (\mathbf{E} \times \mathbf{b})$