Page 237 - Computational Modeling in Biomedical Engineering and Medical Physics

226 Computational Modeling in Biomedical Engineering and Medical Physics

Figure 7.3 Geometric characteristics of the elementary circular turn of a stimulating coil (Morega, 2000).
(A) α 5 0, theturnisinthe (xOy, z 5 0) plane. (B) The turn can rotate by 0 # α # π/2, with A as a fix
point.

From the geometry shown in Fig. 7.3, the equations connecting the variables of the
geometry described in Fig. 7.2 to the new variables (angles α and ϕ)are givenasfollows:

p ffiffiffi
OA 5 r 2 2 1 ; ð7:9aÞ

r p ffiffiffi
x 0 5 p ffiffiffi 2 2 1 1 cos α 1 sin ϕ 2 cos α cos ϕ ; ð7:9bÞ
2

r p ffiffiffi
y 0 5 p ffiffiffi 2 2 1 1 cos α 2 sin ϕ 2 cos α cos ϕ ; ð7:9cÞ
2

z 0 5 r sin α 1 2 cos ϕð Þ: ð7:9dÞ

dl 5 dl x i 1 dl y j 1 dl z k 5 dx 0 i 1 dy 0 j 1 dz 0 k
2 3
r r ð7:9eÞ
5 p cos ϕ 1 cos α sin ϕð Þi 1 p ffiffiffi 2cos ϕ 1 cos α sin ϕð Þj 1 r sin α sin ϕ k dϕ:
4
5
ffiffiffi
2 2
With the new variables, Eq. (7.8) becomes
! (
μ di x 2 x 0 cosϕ 1 cosαsinϕ
@E x 0
d 5 r p ffiffiffi
@x 4π dt R 3 2
ð7:10Þ
" ! # )
2 2 2
ð
ð x2x 0 Þ 2 y2y 0 Þ z 2 z 0 ð x2x 0 Þ z 2 z 0 Þ
ð
1 1 1 1 sinαsinϕ dϕ;
ρ 4 R ρ R 3
2