3.4: Quasistatic Force Density

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Page ID: 28134

James R. Melcher
Massachusetts Institute of Technology via MIT OpenCourseWare

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The Lorentz force density, Equation 3.2.2, is composed of what will be termed, respectively, an electric force density and a magnetic force density

\[ \overrightarrow{F} = \rho_f \overrightarrow{E} + \overrightarrow{J}_f \times \mu_o \overrightarrow{H} \label{1} \]

It is found in a wide range of applications that the force density is predominantly one or the other of these contributions. Polarization and magnetization force densities, not included in Equation \ref{1}, are similarly identified with the respective quasistatic systems. In this section, dimensional arguments are given that demonstrate that the electric force density generally dominates in electroquasistatic systems, while the magnetic force density dominates in magnetoquasistatic systems.

The line of reasoning is an extension of that introduced in Sec. 2.2. The force density is normalized in accordance with Equation 2.3.4 and the free current density is represented as having the form of Equation 2.3.1. Thus,

\[ \overrightarrow{F} = \frac{\varepsilon_o \mathscr{E}^2}{l} [ \rho_f \overrightarrow{E} + \frac{\tau_m}{\tau} (\sigma \overrightarrow{E} + \frac{\tau_e}{\tau} \overrightarrow{J}_v ) \times \overrightarrow{H} ] \quad \text{EQS} \label{2} \]

\[ \overrightarrow{F} = \frac{\mu_o \mathscr{H}^2}{l} [ (\frac{\tau_{em}}{\tau})^2 \rho_f \overrightarrow{E} + (\frac{\tau_m}{\tau} \overrightarrow{E} + \overrightarrow{J}_v ) \times \overrightarrow{H} ] \quad \text{MQS} \label{3} \]

The relative values of the time constants are summarized by Figure 2.3.1. In the electroquasistatic system, \(\tau_m/ \tau << 1\) and \( \tau_m \tau_e / \tau^2 = (\tau_{em} / \tau)^2 << 1\). Hence, the free charge density term is zero-order in Equation \ref{1}, and the magnetic term is consistently ignored\(^1\) In the magnetoquasistatic force density of Equation \ref{3}, \((\tau_{em}/\tau)^2 << 1\), and the free charge force density is negligible compared to the magnetic term. Hence, the second term of Equation \ref{1} is used to the exclusion of the first in magnetoquasistatic systems.