Page 20 - Computational Modeling in Biomedical Engineering and Medical Physics

6 Computational Modeling in Biomedical Engineering and Medical Physics

field (EMF) whose only interaction with its environment is of electromagnetic nature
the work interaction at the boundary is as follows (Mocanu, 1981):
I I ð ð
@D @B
ð E 3 HÞn i dA 5 Sn i dA 5 EJdv 1 E 1 H dv : ð1:12Þ
Σ Σ V Σ V Σ @t @t
|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl}
|fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
_ W Σ;em _ W Σ;em _ Q dE em
dt
This integral relation states the instantaneous balance between the total electromagnetic
_
, the total internal thermal power (heat rate) by Joule effect, Q,and the
work rate, _ W Σ; em
2
total change in internal electromagnetic energy, dE em .In Eq. (1.12), D [C/m ]isthe electric
dt
2
flux density, S [VA/m ]is Poynting’s vector, and n i is the interior normal to the boundary,
Σ. It is worth noting that the electromagnetic energy is actually the sum of two distinct
contributions, presented usually as the electrical energy density (only electric field quantities),
e el 5 DE , and magnetic energy density (only magnetic field quantities), e mag 5 BH .These
2 2
forms are of importance in EMF problems as their gradients provide for the electric and
magnetic body forces (Chapter 6: Magnetic Drug Targeting).
Eq. (1.12) indicates two EMF-produced heat sources for inclusion in the first law energy
balance: an Ohmic heat source related to the electrical conduction (by Joule effect) and a
Hertzian heat source, related to the displacement electrical current (by dielectric heating)
@D @ 1 @E 2 @P
p Joule 5 EJ; p dielectric 5 E 5 E ð ε 0 E 1 PÞ 5 ε 0 1 E ; ð1:13Þ
@t @t 2 @t @t
2
where P [C/m ] is the electrical polarization.
In nonlinear dielectric media with hysteresis, the per-cycle integral of p dielectric for a cyclic
H
excitation EdP 5 Q dielectric;cycle is the heat released per cycle through polariza-
Cycle
tion depolarization (Warburg theorem; Warburg, 1881). A similar discussion shows off a sim-
H
ilar heat source for magnetic nonlinear media with hysteresis, HdB 5 Q magnetic;cycle ,which
Cycle
occurs in microwave (MW) hyperthermia [Chapter 8: Hyperthermia and Ablation
(Thermotherapy Methods)].
For a harmonic excitation, with ω 5 2πf (f is frequency), the simplified complex repre-
sentation of the local form of the power balance Eq. (1.12) is as follows (Mocanu, 1981):

2 div SðÞ 52 divðE 3 H Þ 5 j ωμHH 1 σEE 2 j ωεEE ð1:14Þ
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflffl{zfflffl} |fflfflffl{zfflfflffl}
S elmag P mag P J P Hertz
5 jωμHH 1 σ 2 jωεÞ EE ;

ð
|fflfflfflfflffl{zfflfflfflfflffl}
ε 0