Pre-Quantum Electrodynamics
Steady Currentsems.ms.lf.sc
- PM 4.2, 6.1-2
- Gr 5.1.3
Consider an infinitesimal surface \(\Delta {\cal S}\) with normal unit vector \(\hat{\bf n}\).
In the region containing this surface, let there be a flow of charge described by a charge density \(\rho\) moving at velocity \({\bf v}\).
The charge flowing through \(\Delta {\cal S}\) in one unit of time is thus \(\Delta q = {\bf J} \cdot \hat{\bf n} ~\Delta {\cal S} ~\Delta t\).
This leads us to define a current density \({\bf J}\) as the vector representing the charge flowing through a unit area per unit time, with direction along the motion of the charges. Then, \(\Delta q = \rho {\bf v} \cdot \hat{\bf n} ~\Delta {\cal S} ~\Delta t\) so we can identify
\[ {\bf J} = \rho {\bf v} \tag{vcurd}\label{vcurd} \]
Similarly, a current running down a wire can be described by a linear charge density \(\lambda\) moving at velocity \({\bf v}\), giving the current
\[ {\bf I} = \lambda {\bf v} \tag{lcurd}\label{lcur} \]
For charge flowing on a surface, we can describe the surface current density as a surface charge density \(\sigma\) moving at velocity \({\bf v}\),
\[ {\bf K} = \sigma {\bf v} \tag{scurd}\label{scurd} \]
Magnetic forces on wire, surface and volume carrying current densities:
\[ {\bf F}_{mag} = \int {\bf v} \times {\bf B} ~\lambda dl = \int |{\bf I}| ~d{\bf l} \times {\bf B} \tag{Fmagl}\label{Fmagl} \]
\[ {\bf F}_{mag} = \int {\bf v} \times {\bf B} ~\sigma da = \int {\bf K} \times {\bf B} ~da \tag{Fmags}\label{Fmags} \]
\[ {\bf F}_{mag} = \int {\bf v} \times {\bf B} ~\rho d\tau = \int {\bf J} \times {\bf B} ~d\tau \tag{Fmagv}\label{Fmagv} \]

Created: 2024-02-27 Tue 10:31