##### Steady Currents
ems.ms.lf.sc

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 \(\Delta q = {\bf J} \cdot {\bf n} ~\Delta {\cal S} ~\Delta t\).

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, \(\Delta q = \rho {\bf v} \cdot {\bf n} ~\Delta {\cal S} ~\Delta t\) so we can identify

- Gr (5.26)
- PM (4.4)

\[ {\bf J} = \rho {\bf v} \tag{vcurd}\label{vcurd} \]

Similarly, a current running down a wire can be described by a linear charge density \(\lambda\) moving at velocity \({\bf v}\), giving the current

- Gr (5.14)

\[ {\bf I} = \lambda {\bf v} \tag{lcurd}\label{lcur} \]

For charge flowing on a surface, we can describe the surface current density as a surface charge density \(\sigma\) moving at velocity \({\bf v}\),

- Gr (5.23)

\[ {\bf K} = \sigma {\bf v} \tag{scurd}\label{scurd} \]

Magnetic forces on wire, surface and volume carrying current densities:

- Gr (5.16)

\[ {\bf F}_{mag} = \int {\bf v} \times {\bf B} ~\lambda dl = \int |{\bf I}| ~d{\bf l} \times {\bf B} \tag{Fmagl}\label{Fmagl} \]

- Gr (5.24)

\[ {\bf F}_{mag} = \int {\bf v} \times {\bf B} ~\sigma da = \int {\bf K} \times {\bf B} ~da \tag{Fmags}\label{Fmags} \]

- Gr (5.27)

\[ {\bf F}_{mag} = \int {\bf v} \times {\bf B} ~\rho d\tau = \int {\bf J} \times {\bf B} ~d\tau \tag{Fmagv}\label{Fmagv} \]

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42