ELECTROMAGNETIC FUNDAMENTALS FOR SIGNAL INTEGRITY - Advanced Signal Integrity for High-Speed Digital Design - page 53

which can be simplified:

1
2 µ 0 all space B 2 dV

W m = (2-106)

This leads to the definition of the volume energy density , which expresses the
stored energy in a magnetic field:

B 2
2 µ 0

joules / m 3

w m = (2-107)

2.6 POWER FLOW AND THE POYNTING VECTOR

As electromagnetic waves propagate through space, they carry power. The goal
of this section is to derive a relationship between the electric and the magnetic
field vectors and the power transferred. To begin this derivation, the energy stored
in both the electric and magnetic fields must be quantified within a volume of
space. Consider Figure 2-22, which depicts a cubic volume of space with an
electromagnetic plane wave propagating through it. To quantify the total power,
which is the transfer of energy over time, propagating through the cube, all
sources of energy must be accounted for. The power balance equation can be
expressed as

P A = P S − P L − P EM (2-108)

where P A is the power flowing through the surface of the far end of the cube of
space with area A , as shown in Figure 2-22, P S represents any sources of power
within the cube, P L represents the losses within the cube that dissipate power as
heat (such as resistive losses), and P EM represent the power contained within the
electromagnetic waves that propagate into the cube. If we assume a source-free
and loss-free medium, the power flowing into the cube must equal the power
flowing out of the cube, where the minus sign convention is chosen because the
power is flowing out of the surface:

P A =− P EM (2-109)

x

z 1 z 2

Power out

Power in

P EM

P A

z

y

Surface area = A

Figure 2-22 Volume of space used to calculate power flow and the Poynting vector.