Magnetostatics

Moving charges produce electric current and electric currents produce magnetic field. We shall now consider the relationship between constant ( time independent ) currents and magnetic fields. Consider a charge distribution $\rho(\vec r) = \sum_i q_i \delta(\vec r - \vec r_i)$ consisting of a number of particles having charge $q_i$. If the charges are not stationary, the electric current is defined as

$$\vec j(\vec r) = \sum_i q_i \frac{d \vec r_i}{dt} \delta(\vec r - \vec r_i)$$

. The current and charge density together satisfy continuity equation

$$\vec \nabla \cdot \vec j(\vec r) + \frac{\partial \rho(\vec r)}{\partial t} = 0.$$ (88)

This equation follows from conservation of charge. If we consider a volume, the rate of change in the charge in this volume is given by the rate at which the charge flows out of the surface of the volume. The continuity equation above is the differential form of charge conservation.