1.7 Complex EM Properties

For many geophysical applications we neglect the permittivity (ε or K) term and assume the magnetic permeability is that of free space, i.e., K_m = 1. This cannot be done for ground penetrating radar, GPR. In this case we must treat the total current density, J_T as

JT = JC +∂D/∂t.

(1.7.1)

Here, J_C is the same current density, J defined earlier, however, in the context here it is usually referred to as the conduction current density to distinguish it from the displacement current density, ∂D/∂t.

By assuming that the time dependency of all fields is e^+iωt we see that

JT = JC + iωD	(1.7.2)
= (σ + iωε)E.	(1.7.3)

(Using the exponential, e^+iωt to express a sinusoidal time variation with time is discussed in Appendix B). Equation 1.7.3 allows a generalized complex expression of Ohm's law (equation 1.5.1). This uses the definition of complex conductivity, σ* as

σ* = σ' + iσ''

(1.7.4)

and yields

JT = σ*E,

(1.7.5)

 

where σ' = σ and σ'' = ωε are the real and imaginary parts of the complex conductivity, σ* respectively. A convenient way to express the relative amount of conduction current versus displacement current is the loss tangent, tan δ.

tan δ = σ / ωε = conduction current density displacement current density

(1.7.6)

Another way to express J_T is to modify the J_T equation 1.7.3 above as

JT	= iω(σ/ iω + e)E
	= iω(ε - iσ/ ω)E
	= iωε*E	(1.7.7)

This leads to the definition of complex electric permittivity, ε* (von Hippel, 1954) as

ε * = ε' – iε'' ,

(1.7.8)

where ε' and ε'' are the real and imaginary parts of the complex permittivity, ε*, respectively. This allows the alternate expression of the loss tangent to be

tan δ = ε'' / ε'

(1.7.9)

The complex dielectric constant, K* follows as

K*	= ε*/ε0
	= K' – iK''.	(1.7.10)

Davis and Annan (1989) prefer to separate the imaginary part of the dielectric constant into two parts: 1) a dc component of conductivity σ_dc and 2) a high frequency contribution, K''. Then

K*	= K' – iK'''
	= K' + i(K'' + σdc/ωε0)	(1.7.11)

The relations above can be represented in the complex plane. Figure 1.9 a, b, and c contain such examples.

a)	b)	c)
Figure 1.9 Complex graphical representations of a) total current density, JT, b) conductivity, σ*, and c) dielectric constant, K*.