Page 37 - Maxwell House

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS 17

Δ Δ � x (1.12)
= + �
∆ ∆ ∆
According to (1.5) and (1.6) we obtain

3
= + x [N/m ] (1.13)

Here = lim ( ⁄ ∆) is the macroscopic volume density of force at point O. Eventually,

∆→0
equations (1.11) and (1.13) can be used to measure a given electromagnetic field by observing
the motion of charged particles.
According to (1.11) the total exerted force consists of two parts. The first term = Δ

is called the electric force, while the second one = Δ x is deduced as the magnetic

force. Eventually, in the bounds of macroscopic electrodynamics the limit should exist
lim ( ⁄ Δ ) = + x [V/m] (1.14)

Δ →0
As we have demonstrated above in Example #2 of Section 1.3.3, both terms on the right side of
(1.14) have the same unit dimension [V/m]. It tells us that the second vectorial term is must be
some kind of the electric field
= x (1.15)

induced by magnetic B field. This remarkable fact that the time-varying magnetic field can
cause the electric field was discovered by British scientist Michael Faraday in 1831.

We will return to Lorentz’s equation later. Now, let us move to the construction business of
building, so called, House of Maxwell’s Electrodynamics. [4]

1.5.3 House of Maxwell’s Electrodynamics (Maxwell’s House)
Loosely speaking, our Maxwell’s House is no more than the classification diagram
summarizing Maxwell’s equations graphically, helping to memorize and visualizing them as
the items in "memory palace." We intend to translate Maxwell’s equations into images that are
then placed on the palace walls, floor, attic, and basement. So we can mentally navigate
ourselves through that space and pick up those images we left there and translate them back to
what we memorized. Fully furnished Maxwell’s House [4] is shown in Figure 1.5.1a and b. The
one can read off each of Maxwell’s equations in differential form by just adding all incoming
through the black arrow quantities at each node and setting it equal to the amount at the node.
For the sake of drawing simplicity, we denote the operator by the symbol from Table 1.6

≡ (1.16)

and omitted the front symbol ∆ and subscript e for charge q in Figure 1.5.1 to make the picture
more readable. It turns out that Maxwell’s House consists of three levels: basement, living
room, and attic. The left wall in the living room of 1.5.1a can be called as Faraday-Lorentz’s
wall, the right wall of the same room can be called Ampere-Maxwell’s wall, the attic belongs
to electric charges and currents, while the magnetic charges and currents go to the basement as
not found yet but widely used in computational electrodynamics. It is worth to note that in this
diagram the vectors , E and B defining the force exerted by electromagnetic fields are
located on the Faraday-Lorentz’s wall while all the derivative vectors D and H flock on the
Ampere-Maxwell’s wall. In the case of electrostatic and steady magnetic fields the time