Page 47 - Maxwell House

BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS 27

= ∯ ∘ = − (1.33)

is the integral form of the continuity equation. But remember that this equation describes the
movement of positive charges. If the current is defined as the flow
0 of electrons, in (1.33) must be replaced by − . The deferential

form of the continuity equation follows from (1.33) if we transform
(see Appendix) the surface integral to volume one applying (1.30)
q to the current density as
qv -
∯ ∘ = ∫ ∘ (1.34)

Replacing in (1.33) = ∫ and grouping the terms we

obtain
Figure 1.6.7 � = 0 (1.35)
Populated House of ∫ � ∘ +

Maxwell’s Pronouncing the “magic” words, we finally get the continuity
Electrodynamics equation in differential form

∘ + = 0 (1.36)

Now, we can populate Maxwell’s House attic with new residents , and , as shown in

0

Figure 1.6.7.
st
1.6.11 Lorentz’s Force Equation and 1 Maxwell’s Equation
Looking back at Lorentz force equation (1.11) we note that the magnetic field exerts the force
on a moving charge that can be described through the equivalent electric field (1.15)

as = x . That is not something occasional. The most remarkable fact is that magnetic

and electric fields are in reality just different aspects of the same and inseparable phenomenon
— the electromagnetic force. Imagine that a girl shown in Figure 1.6.6a is traveling on a train
car touching a charged metal sphere and having a bunch of tools including her hair to measure
electric and magnetic fields. Taking into account that the girl, the sphere, and all her instruments
are at rest relative to each other the girl’s instruments can detect only electrical fields.
Nevertheless, her friend standing with an identical set of tools on the platform while the train
passes, will detect not only electric fields but magnetic fields too, because for the friend and
his/her instruments the charged sphere in motion is equivalent to some current and the source
of magnetic field. Eventually, nothing changes if the train stops and the girl’s friend starts
running nearby the train car.

We will use such duality or principle of relativity sometimes implicitly to bridge the gap
between static and time-varying fields, static and moving charges, electrical and magnetic
fields, et cetera. Electric and magnetic fields are just two faces of the same natural phenomenon
like the ancient Roman two-faced god Janus, who looks simultaneously to the future and the
past, symbolizing the transition from one condition to another . More details about such duality
10
effects can be found in the special theory of relativity and are beyond the scope of this book.

10 It looks like Romans believing in such god foresaw electromagnetic field propagation long ahead of
Maxwell.