Page 54 - Maxwell House

34 Chapter 1

Eventually, in (1.63) both time and space coordinates are independent variables. If so,
differentiation with respect to time or coordinates can be done in any order, time first then
coordinate (divergence operator) and vice versa. According to (1.62), = and
⁄

∘ + ∘ = ∘ ( + ) = 0 (1.64)

Therefore, the divergence of the net current = + at any point of space and any

moment of time is always equal to zero.
To illustrate and understand this result let us use the line of force that offers, as we have
displayed before, quite a good visualization of vectorial electrical fields and, eventually, the
same approach can be used for any type of vectorial fields including the net current .

Let us wrap some volume V (all points inside V) in a closed surface area A (all points on the
boundary of V), as shown in Figure 1.6.14, and count the
number of force lines crossing the surface inward and outward.
More force lines leave than entering means the positive
divergence and presence of some additional EM field sources
inside the volume V. Alternatively, the divergence is negative
when more force lines go into the volume then go away from
Figure 1.6.14 Zero it meaning that there is some sink inside the volume V
divergence illustration absorbing part of EM energy. If so, the zero divergence means
nothing is lost or added inside V and the number of the exiting
and leaving lines of force are equal, as Figure 1.6.14 illustrates. It means that if for some reason
the electric current decreases at some point in V the displacement current increases at

precisely at the same rate at this point and vice versa.
Therefore, these two currents are unbreakable (except in the case of static (time-undependable)
nd
⁄
electric fields when = ≡ 0). If so, 2 Maxwell’s
equation in integral form must take the following view

∮ ∘ = ∬ ∘ + (1.65)

Let us discuss shortly the well-known Ampere’s law following from
this equation. In the case of static electric field, ≡ 0 and
⁄
Figure 1.6.15 ∮ ∘ = (1.66)
Magnetic field
around long This equation describes the magnetic field wounding around the
straight wire with linear wire carrying the steady electric current (see Figure 1.6.15). It
current was discovered in the 1820s by French scientist Andre-Marie
Ampere. The great usefulness of Ampere’s law is not just like the
tool to calculate the magnetic field created by steady currents and vice versa, but it opens the
possibility to define and measure the magnetic field strength very naturally. Assume that in
(1.66) the circular loop L of radius R wraps around a long thin straight wire carrying the
current . Figure 1.6.15 illustrates how to apply Right-Hand Rule (RHR) to get the magnetic

field direction without the corkscrew “opening the magnetic bottle”: point your thumb in the
direction of current and curl your fingers into a half-circle around the wire. Then the fingers