Pre-Quantum Electrodynamics

Consider a little surface \(\Delta {\cal S}\) having a normal unit vector \(\hat{\bf n}\).

We start by defining a {\bf current density} \({\bf J}\) as the vector representing the amount of charge flowing through a unit area per unit time. Its direction is along the motion of the charges.

The charge flowing through \(\Delta {\cal S}\) in one unit of time is thus \(\Delta q = {\bf J} \cdot {\bf n} ~\Delta {\cal S} ~\Delta t\).

Let the flow of charge be given by a charge density \(\rho\) moving at velocity \({\bf v}\). Then, \(\Delta q = \rho {\bf v} \cdot {\bf n} ~\Delta {\cal S} ~\Delta t\) so we can identify \[ {\bf J} = \rho {\bf v} \label{Gr(5.26)} \] The total current \(I\) going through a surface \({\cal S}\) is \[ I = \int_{\cal S} {\bf J} \cdot d{\bf a} \label{Gr(5.28)} \] Over a closed surface, we can use the divergence theorem, \[ \oint_{\cal S} {\bf J} \cdot d{\bf a} = \int_{\cal V} d\tau {\boldsymbol \nabla} \cdot {\bf J} \] Since charge is conserved, \[ \int_{\cal V} d\tau {\boldsymbol \nabla} \cdot {\bf J} = -\frac{d}{dt} \int_{\cal V} d\tau \rho = - \int_{\cal V} d\tau \frac{\partial \rho}{\partial t}. \] Since this is valid for any volume, we get the

Current down a wire: linear charge density \(\lambda\) moving at velocity \({\bf v}\) means current \[ {\bf I} = \lambda {\bf v} \label{Gr(5.14)} \] Current on a surface: {\bf surface current density} made up of surface charge density \(\sigma\) moving at velocity \({\bf v}\) \[ {\bf K} = \sigma {\bf v} \label{Gr(5.23)} \]