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POYNTING's THEOREM 129

a) b) c)

Figure 3.1.12 a) Solenoid with core, b) Symbol of inductance with core, c) Symbol of pure
inductance (no core).
The ratio

2
2 −1 2
ℒ = = [(L m���) (m /m) = L] (3.31)
2 0
ℎ

is called inductance, measured in Henry (see Chapter 1, Table 1.5), and characterized the ability
of any circuit device to store the energy in the form of magnetic fields. The ferromagnetic core
concentrating the inductor’s magnetic field inside the core increases the inductance times.

The schematic symbol for inductor with and without ferromagnetic core is shown in Figure
3.1.12b, and c, respectively. In fact, the expression (3.31) is some approximation because we
ignored the solenoid external magnetic fields. Taking into consideration Faraday’s law in the
form (1.42) and calculating the magnetic flux we can get one more equation of the circuit theory

(3.32)
() = −ℒ

defining the drop potential at the inductor.

3.1.13 Parasitic Parameters

Introducing the parallel plate capacitor, we suggested that only the electric fields carry the total
stored energy. Now it is time to ask how good such approximation because the time dependable

()
electrical fields in the form of displacement current = ∯ ∘ (defined by (1.61)) are

the source of alternative magnetic fields shown in Figure 3.1.10a by blue lines. In the prior
section we demonstrated that any excessive concentration energy in magnetic fields means the
existence of the “unwanted” or parasitic inductance. Plugging (3.24) in the second Maxwell’s
equation (line 2 in Table 1.7) and evaluating the linear and surface integrals we have

(, ) = (3.33)

2
Then according to (3.33) the magnetic storage is
4 2

= ( ) 2 � � (3.34)
0

16
Using the fact that in Figure 3.1.10a the electric current carried by the wires connecting the
capacitor to the source of alternative voltage is equal to ( = )
2

2

⁄
= = ∯ ∂ ∘ = (3.35)