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126 Chapter 3

3.1.10 Poynting’s Theorem and Circuit Analysis

Now, let us illustrate with simple examples how Pointing’s theorem works. From the electric
circuit course, we know that the inductor, capacitors, and resistors (introduced in Chapter 2) are
the fundamental lumped elements of digital and analog circuits. Historically all these circuitry
elements came from low-frequency applications. Then step by step the equivalent circuit
presentations based on them expanded at much higher frequencies and became practically
useful for analysis at any frequencies up to the frequencies of the optical spectrum.

The intimate relationship between the EM stored or dissipated energy, and the lumped circuitry
element parameters lead to a much better understanding of circuit theory as well as Maxwell’s
equations. Poynting’s theorem tells us that any physical element or any part of it where the
stored electrical energy much exceeds the magnetic one behaves mostly as a capacitor. The
element with a contrariwise combination of energy acts primarily as an inductance. Moreover,
from Maxwell’s equations follow that the instantaneous electric fields are the source of
instantaneous magnetic fields and vs. making electric and magnetic fields are almost
inseparable. Therefore, there are no real inductors, capacitors, or resistors at any frequency
except DC. For example, in the case of time-varying fields the correct equivalent circuit of any
capacitor must include small but nonzero inductor and resistor, etc.

Multiple numerical methods of complex electromagnetic problems solution are based on
analysis of an array of three-dimensional lumped elements interconnected in a multiport
network. Typically, the lumped element values for each network node are calculated using the
Poynting’s theorem. The resistivity, capacitance and inductance concept built on the energy
conservation law gives us the opportunity to develop efficient numerical techniques for
Maxwell’s equations solution breaking complicated structures into essential small parts (spatial
discretization). Building the equivalent circuit of material bodies and surrounding them free
space is extremely general and requires the knowledge of only electrical/magnetic fields and
medium parameters , , .

We are going to discuss this matter detailly in following chapters. Now, let illustrate only the
potent application of the energy concept following from Poynting’s theorem for circuit elements
as a capacitor, inductance and the effect of mutual coupling between circuit elements. Keep in
mind that all results below will be based on the quasistatic approximation of electric potential
and current.

3.1.11 Concept of Capacitance

We start by considering a charged parallel plate capacitor shown in Figure 3.1.10a. This
capacitor consists of two parallel metallic disks of equal area = at the distance d. Each
2
disk carries a charge of magnitude Q while a dielectric with constant occupies the gap and

entire space outside. When the gap ≪ the most part of E-fields is evidently concetrated
between the plates, and only the small energy portion settles in the exterior or fringe electric
fields (see Figure 3.1.10b). If so, the electric field distribution between plates can be asumed as
close to uniform that Figure 3.1.10a dermonstrates.