1.5: Models and Approximations

Last updated
Save as PDF

Page ID: 28121

James R. Melcher
Massachusetts Institute of Technology via MIT OpenCourseWare

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

There are three classes of approximation, used repeatedly in the following chapters, that should be recognized as a recurring theme. Formally, these are based on time-rate, space-rate and amplitude parameter expansions of the relevant laws.

The time-rate approximation gives rise to a quasistatic model, and exploits the fact that temporal rates of change of interest are slow compared to one or more times characterizing certain dynamical processes. Some possible times are given in Table 1.4.1. Both for electroquasistatics and magnetoquasistatics, the critical time is the electromagnetic wave transit time, \(\tau_{em}\) (Sec. 2.3).

Space-rate approximations lead to quasi-one-dimensional (or two-dimensional) models. These are also known as long-wave models. Here, fields or deformations in a "transverse" direction can be approximated as being slowly varying with respect to a "longitudinal" direction. The magnetic field in a narrow but spatially varying air gap and the flow of a gas through a duct of slowly varying cross section are examples.

Amplitude parameter expansions carried to first order result in linearized models. Often they are used to describe dynamics departing from a static or steady equilibrium. Long-wave and linearized models are discussed and exemplified in Sec. 4.12, and are otherwise used repeatedly without formality.