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3 Fundamentals of the Formation of Biosignals

However, they are attenuated on their way there and also overlap with the fields of

ions from other points of origin.

In general, Maxwell’s equations [18, 47, 61] also apply here:

∇× H = ϵ ^∂E

∂t ^{+ J ,}

(3.3)

∇× E = −μ ^∂H

∂t ^,

(3.4)

∇⋅(ϵE) = ρv ,

(3.5)

∇⋅J = −^∂ρv

∂t ^,

(3.6)

∇⋅(μH) = 0 .

(3.7)

In Equation 3.3 to Equation 3.7, E and H are the electric and magnetic field strengths,

J is the current density, ϵ and μ are the electric and magnetic permeabilities, ρv is

the space charge density and ∇is the Nabla operator (∇:=

∂

∂s ^{= ex ∂}

∂x ^{+ ey ∂}

∂y ^{+ ez ∂}

∂z

for Cartesian coordinates). These equations can be simplified and reformulated for

medical considerations:

–

The electrical and magnetic permeabilities are the same as those of the vacuum,

i.e. ϵ = ϵ0 = 8.854 ⋅10⁻¹²F/m and μ = μ0 = 4π⋅10

−7H/m.

–

Since the static space charge density ρv is negligible in a conducting system such

as a body, Equation 3.5 need not be considered. Although thereare charged ions on

a cell membrane, they balance each other on the different sides of the membrane,

such as in a plate capacitor.

–

The current density J can be divided into a current passing through the elec-

tric field JE = κE (κ for electrical conductivity), and an ionic current that flows

between the cell membranes. It can be described by an internal current source Ji,

ie. h. J = JE + Ji = κE + Ji.

A further simplification results if the Maxwell equations are subjected to a Fourier

transformation and used in complex form in the image domain. With

E(t) = ¹

2π

∞

∫

−∞

E(ω) e^jωt dω ,

H(t) = ¹

2π

∞

∫

−∞

H(ω) e^jωtd ω

results because of

∂E(t)

∂t

∘−−∙jω E(ω)

but.

∂H(t)

∂t

∘−−∙jω H(ω)

(3.8)