[NOTES/EM-10001]-Overview of Electrodynamics Equations

Important equations of electrodynamics,the equation of continuity, the Lorentz force and the Maxwell's equations are summarized.  

1. Charges and Currents

The current density \(\vec{j}(\vec{r},t)\) and the charge density \(\rho(\vec{r},t)\) obey the equation of continuity \begin{equation}\label{EQ11} \pp[\rho]{r} + \nabla\cdot \vec{j} =0. \end{equation} The charge density \(\rho(\vec{r},t)\) is the charge per unit volume. Let \(V\) be a volume enclosed by a surface \(S\). The total charge in volume \(V\) is given by the volume integral,$\iiint_V\rho d^3r$, of the charge density. The amount of charge flowing out of the volume \(V\) is given by the surface integral of the current density, $\iint_S \vec{j}\cdot \overrightarrow{dS}$, over the surface \(S\). The equation \eqRef{EQ11} represents a local conservation of charge. It says that the rate of increase of total charge enclosed in a volume is equal to the rate at which the charge is flowing into the volume from its boundary. This statement follows from considering the volume integral of EQ11 over \(V\) and applying Gauss divergence theorem.

2.Lorentz force

The force on a charged particle, moving with velocity \(\vec{v}\) in electric field \(\vec{E}\) and magnetic field \(\vec{B}\) is given by \begin{equation} \vec{F} = q( \vec{E} + \vec{v}\times \vec{B}), \end{equation} where \(q\) is the charge on the particle,

3.Maxwell's equations in absence of a medium

The Maxwell's equations, in absence of a dielectric and magnetic media, are as follows. \begin{eqnarray}\label{EQ01} \nabla\cdot \vec{E} &=& \frac{\rho}{\epsilon_0}\\\label{EQ02} \nabla \times \vec{E} &=& -\pp[\vec{B}]{t}\\\label{EQ03} \nabla \cdot \vec{B} &=&0\\\label{EQ04} \nabla \times \vec{B} &=& \frac{\mu_0}{4\pi} \vec{j} + \epsilon_0\mu_0 \pp[\vec{E}]{t}. \end{eqnarray} The charge density and the current density are related by \(\vec{j}= \rho \vec{v}\). To the above equations we add the Lorentz force equation \begin{equation}\label{EQ09} \vec{F} = q (\vec{E} + \vec{v}\times \vec{b}), \end{equation} In the time varying situations both electric and magnetic fields have to be nonzero. We say that time varying electric field produces magnetic field and vice versa. An important relation between the charge density \(\rho\) and the current density \(j\) is the equation of continuity. \begin{equation}\label{EQ10} \pp[\rho]{t} + \nabla \cdot \vec{j} =0. \end{equation} This equation represents conservation of electric charge in local form. The above equations fully describe the electromagnetic phenomena in absence of a medium.