3.2 Electrophysiology of the Heart

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after insertion into Equation 3.3, Equation 3.4 and Equation 3.7 with the following sim-

plifications:

∇× H = (κ + jωϵ0) E + Ji ,

(3.9)

∇× E = −jωμ0 H ,

(3.10)

∇⋅H = 0 .

(3.11)

Since according to Equation 3.11 the divergence of H vanishes, H can be expressed

by the rotation of any scalar vector field A for easier determination of the solution of

these Maxwell equations, after the divergence of a rotation of any vector field always

vanishes. Here one chooses, for example.

μ0 H := ∇× A .

(3.12)

After inserting in Equation 3.10 it follows:

∇× (E + jωA) = 0 .

(3.13)

Since the rotation of E + jωA also vanishes, it can now be expressed by any scalar

function ϕ as follows:

E + jωA = −∇ϕ .

(3.14)

According to the Helmholtz theorem, a vector field is uniquely described by specifying

its rotation and divergence [60]. Since for the vector field A according to Equation 3.12

only the rotation has been defined so far, the divergence would have to be specified

additionally. For this purpose

∇⋅A := −κμ0Φ

(3.15)

can be defined. With the help of this definition, the Maxwell equations can now be re-

duced to the solution of an equation for the vector potential A. If Equation 3.12, Equa-

tion 3.14 and Equation 3.15 are substituted in Equation 3.10 and if one additionally

considers the Graßman development theorem

∇× ∇× A = ∇(∇⋅A) −∇²A ,

(3.16)

then we obtain the vectorial Helmholtz-equation

∇²A −jωμ0κA = −μ0Ji ,

(3.17)

whose solution is well known in classical electromagnetic theory and is given by

A = ^μ0

4π ^∫

Jie^−kr

r

dv

(3.18)

k² = jωκ(1 + jωϵ0/κ)

(k: wave vector)

r² = (x −x^󸀠)² + (y −y^󸀠)² + (z −z^󸀠)²

(r: distance current source to measuring point)