3.2: The Lorentz Force Density

Although macroscopic forces were the first measured in the development of electricity and magnetism, it is now normally accepted that the fundamental force is that on a "test" charge. This charge might be a single electron in free space. If the charged particle has a total charge \(q\) and moves with a velocity \(\overrightarrow{v}_p\), then the Lorentz force acting on the particle supporting the charge is

\[ \overrightarrow{f} = q \overrightarrow{E} + q \overrightarrow{v}_p \times \mu_o \overrightarrow{H} \label{1} \]

This statement, like the electrodynamic laws summarized in Chapter 2, is an empirical one. In most of the areas of continuum electromechanics, it is forces due to many charges that are of interest, and it is therefore appropriate to sum the individual forces of Equation \ref{1} over the charges within a given unit of volume to arrive at the Lorentz force density

\[ \overrightarrow{F} = \rho_f \overrightarrow{E} + \overrightarrow{J}_f \times \mu_o \overrightarrow{H} \label{2} \]

Incremental volumes of interest have dimensions much greater than the characteristic distances between particles. But also, for the average electrical field to have meaning, it must be primarily due to sources external to the differential volume of interest. This ensures that, over an incremental volume , each particle experiences essentially the same electric field. The contribution to the field of the charges within the differential volume is negligible. Similar arguments apply to the magnetic field intensity, which must be produced over a given differential volume largely by currents outside the volume.

Equation \ref{2} represents the force density acting on a ponderable medium if means are available for the force on the particles to be transmitted to the medium. The mechanisms by which this happens are diverse, and implicit to the conduction process. Whether the fundamental carriers are electrons in a metal, holes and electrons in a semiconductor or ions in a liquid or gas, the average motions of fundamental charge carriers are superimposed on random motions. The flights of fundamental carriers are interrupted by collisions with lattice molecules (in a solid) or molecules that are themselves in a Brownian equilibrium (in a liquid or gas) with a frequency that is usually extremely high compared to reciprocal times of interest. These collisions transfer momentum from the fundamental charge carriers to the ponderable medium.

To more fully appreciate the transition from the force acting on fundamental carriers, Equation \ref{1}, to that on a material, Equation \ref{2}, it is helpful to make a formal derivation. Although the discussion leads to rather general conclusions, only two families of carriers are now considered, one positive with charge per particle \(q_{+}\) and number density \(n_{+}\) and the other negative with a magnitude of charge \(q_{-}\) and number density \(n_{-}\). The average Lorentz force, Equation \ref{1}, is in equilibrium with an average force representing the effect of collisions on the net migration of the particles:

\[ \begin{align} q_{+} \overrightarrow{E} + q_{+} ( \overrightarrow{v}_{+} + \overrightarrow{v}) \times \mu_o \overrightarrow{H} &= m_{+} \nu_{+} \overrightarrow{v}_{+} \nonumber \\ -q_{-} \overrightarrow{E} - q_{-}(\overrightarrow{v}_{-} + \overrightarrow{v}) \times \mu_o \overrightarrow{H} &= m_{-} \nu_{-} \overrightarrow{v}_{-} \nonumber \end{align} \label{3} \]

The retarding forces on the right are much as would be conceived for a swarm of macroscopic particles moving through a viscous liquid. The average carrier velocities \(\overrightarrow{v}_{\pm}\) are measured relative to the medium which itself has the velocity \(\overrightarrow{v}\). Hence, on the right it is relative velocities of particles and medium that appear, while in the Lorentz force it is total particle velocities that are appropriate. The coefficients for the collisional forces are written as the product of the particle masses \(\overrightarrow{m}_{\pm}\) and collision frequencies \(\nu_{\pm}\) as a matter of convention. Note that the inertial force on the carriers is ignored compared to that due to collisions.This approximation would be invalidated in a plasma if the frequency of an applied electric field intensity were extremely high. But, in many conductors and certainly in the most usual electromechanical situations, the inertial effects of the charge carriers can be ignored (see Problem 3.3.1.)

The charge density and current density are written in terms of the microscopic variables as

\[ \rho_f = n_{+} q_{+} - n_{-} q_{-} \label{4} \]
\[ \begin{align} \overrightarrow{J}_f &= n_{+} q_{+} (\overrightarrow{v}_{+} + \overrightarrow{v}) - n_{-} q_{-} (\overrightarrow{v}_{-} + \overrightarrow{v}) \nonumber \\ &= n_{+} q_{+} \overrightarrow{v}_{+} - n_{-} q_{-} \overrightarrow{v}_{-} + \rho_f \overrightarrow{v} \nonumber \end{align} \label{5} \]

The average force density acting on the ponderable medium is the sum of the right-hand sides of Equation \ref{3}, respectively, multiplied by the particle densities \(n_{\pm}\):
\[ \overrightarrow{F} = n_{+} m_{+} \nu_{+} \overrightarrow{v}_{+} + n_{-} m_{-} \nu_{-} \overrightarrow{v}_{-} \label{6} \]

The point in writing this equation is to formalize the statement that, through some collisional process, the force on the fundamental carriers becomes the force on the medium. It is evident from the next step that, at least in so far as the Lorentz force density is concerned, the details of the collisional equilibrium are not important. The left-hand sides of Equation \ref{3} (regardless, for example, of whether \(m_{\pm} \nu_{\pm}\) are functions of \(v_{\pm})\ or are constant) are substituted for the respective terms in Equation \ref{6} to obtain

\[ \overrightarrow{F} = (n_{+} q_{+} - n_{-} q_{-}) \overrightarrow{E} + [ ( n_{+} q_{+} \overrightarrow{v}_{+} - n_{-} q_{-} \overrightarrow{v}_{-}) + (n_{+} q_{+} - n_{-} q_{-}) \overrightarrow{v} ] \times \mu_o \overrightarrow{H} \label{7} \]

In view of the definitions given by Eqs. \ref{4} and \ref{5}, this expression is the Lorentz force density of Equation\ref{2}. Its validity hinges on there being an instantaneous equilibrium between the forces on the fundamental carriers and the "collisions" with the ponderable medium, but not on the details of that interaction.