Page 56 - Maxwell House

36 Chapter 1

Integral Form Differential Form Comments

6 = = Constitutive relation

7 = 0 = 0 Constitutive relation
A quick look at Table 1.8 reveals that

1. All vector and scalar values are time-independent.
2. The source of electric fields is static electric charges and steady magnetic current
(equivalent or real if their existence will be proved).
3. The source of magnetic fields is static magnetic charges (true if their existence will be
established) and steady electric currents .

4. The displacement current as a source of magnetic fields does not exist anymore. Electric
and magnetic fields are completely decoupled and can be analyzed independently.
5. According to the first equation in Table 1.8, the total work done along any closed loop L
of any shape in the static electric fields is zero. In other words, the static electrical fields
are conservative and the total work performed in such fields is path-independent.
6. According to the second equation in Table 1.8, the total work done along any closed loop
L of any shape in the static magnetic fields is not zero if currents cross this loop. In other
words, the static magnetic fields are not conservative and the total work performed in such
fields is path-dependent.

1.6.17 Electric and Magnetic Energy

Now we ready to make the final step and
introduce the concept of the energy of electric and
magnetic field that is free of the test sensor. Such
makes these fields more measurable and thereby
real. Look back at the expressions (1.21).
Evidently, the potential energy that is
accommodated in E-field and that could be
released as the test charge start moving is equal to
∆ = ∆ (1.71)

Assume that the test charge ∆ is continuous and
spread within the tiny parallelepiped ∆ on the
bounding surface A of some volume V as Figure
1.6.17 demonstrates while the large multifaceted
volume V encloses all of the internal charges
creating the potential within ∆. We see from
Figure 1.6.17 Volume V with source (1.21) and (1.26) that = ∘ = ∘ �∆

charges inside and test charge outside
and ∆ = ∘ ∆ = ∘ �∆. The latter follows
from the fact that ∆ is the only free charge
inside ∆. Consequently, ∆ = ∆∆ and (1.71) yields

∆ = ∘ ∆ (1.72)