Page 36 - Maxwell House

16 Chapter 1

1.5 HOUSE OF MAXWELL’S ELECTRODYNAMICS

1.5.1 Introduction

Let us move to the central goal of this chapter obtaining the set of macroscopic Maxwell’s
equations first in a vacuum and later in materials and making them understandable, without
unnecessary complexity. Before turning to such a task, we would like to notice that there are
many different ways to carry it out but all of them are based on a particular set of axioms. The
central question is how to choose such set. There are three main approaches:
1. Theoretical derivation established on symmetry principles outlined in section 1.2. This
pathway is the most conclusive and can proceed without assuming any results of an
experimental nature. But it is quite tight requiring a good knowledge of high-power
mathematics chapters that are not common in engineering practice.
2. Historical derivation based mainly on experimentally-justified equations and the chain
Coulomb’s law → Ampere’s law → Faraday’s law → Maxwell’s displacement current.
Excellent path but the transition from Coulomb’s and Ampere’s law describing static fields
to time-varying fields is not so understandable and requires plenty of additional
explanations.
3. Engineering derivation based on electric and magnetic charge conservation laws in the
form of Gauss’s law and Lorentz’s force equation. Lorentz’s force equation establishes the
total force exerted by EM field on charged particles and connects as well Maxwell’s
equations to classical mechanics governing particle movements. If so, we can quantify the
force and then the strength of EM field through the measurement of moving particle energy
(a well-established procedure in physics and engineering practice). This fact makes the
mainly invisible electromagnetic fields measurable and applicable from engineer’s
perspective.

We found that the engineering method is the best-suited for our purpose allowing to get the
definition of all fields through the energy they carry. In other words, electric and magnetic fields
become directly accessible to experimental observation. Moreover, it paves the way to
Maxwell’s equations with a minimum of mathematics. We are going to use as much as possible
the intuitive approach and dimensional analysis to avoid unnecessary rigor wherever practicable
in the hope that the readers of our book remember or can quickly refresh the rudiments of EM
field from any introductory physics course [14].

1.5.2 Lorentz’s Force Equation (Axiom #1)
Suppose that EM field could occur in the domain ∆ of free space and we are willing to detect
its existence and then count it at some given point O within ∆. To do so we can put at this
point the sensor #3. In 1892 German scientist Leonard Lorentz established that the combination
of electric E and magnetic B fields exerts the force [12, 17] on this sensor that is equal to

= Δ + Δ x [N] (1.11)

In order to provide the field measurement in the given point only, we have to shrink the
domain ∆ around the point O and take the limit ∆ → 0 in both parts of equation (1.11)