Maxwell S Equations

A good text book for electrodynamics is David Griffith's "Introduction to Electrodynamics".

Basically, there is charge moving around. It creates electric and magnetic fields. These then act back on the charges. This is classical electricity and magnetism. It is summarized by Maxwell's equations:

(1)\begin{align} \nabla \cdot E = \frac{1}{\epsilon_0} \rho \end{align}

(2)
\begin{align} \nabla \times E = -\frac{\partial B}{\partial t} \end{align}

(3)
\begin{align} \nabla \cdot B = 0 \end{align}

(4)
\begin{align} \nabla \times B = \mu_0 J + \mu_0\epsilon_0 \frac{\partial E}{\partial t} \end{align}

For the case of constant currents producing magnetic fields, we have the Biot-Savart law:

(5)\begin{align} B(r) = \frac{\mu_0}{4\pi} \int \frac{K(r')\times \hat {\cal r}}{{\cal r}^2} da' \end{align}