Appendix A
MATHEMATICA CODE (RADIAL CURRENT) - EXAMPLE 2
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mu = 4Pi/10000000;
R1 = 25/1000;
R2 = 35/1000;
l = 4/100;
a = R2/R1;
b = l/R1;
n1 = 100;
r = (a^2 + 2 a Cos[2x] + 1 + b^2)^(1/2);
r0 = (a^2 + 2a Cos[2x] + 1)^(1/2);
r2 = (2a^2 + 2 a^2 Cos[2x] + b^2)^(1/2);
r1 = (2 + 2 Cos[2x] + b^2)^(1/2);
f11 = 2b^3/3/Sin[2x] ArcTan[(a Sin[2x]^2-b^2 Cos[2x])/(b Sin[2x]r)];
f22 = b^3/3/Sin[2x](ArcTan[( Sin[2x]^2-b^2 Cos[2x])/(b Sin[2x]r1)] +
+ ArcTan[(a ^2Sin[2x]^2-b^2 Cos[2x])/(b Sin[2x]r2)]);
T1 = f11-f22;
T2 = 8/3(a^3 + 1) Cos[x]^3 -4/3Cos[x]^2(a^2r2 + r1) + 2/3((a^2 + 1)Cos[2x] + 2a)(r-r0);
T3 = 4b a^2 Cos[x]^2 ArcSinh[b/(2a Cos[x])] + 4b Cos[x]^2 ArcSinh[b/(2Cos[x])]-2b((a^2 + 1)Cos[2x] + 2a)ArcSinh[b/r0];
T4 = 2b^2 (a ArcSinh[(a + a Cos[2x])/(a^2Sin[2x]^2 + b^2)^(1/2)] + ArcSinh[(1 + Cos[2x])/(Sin[2x]^2 + b^2)^(1/2)]);
T5 = -2b^2 (a ArcSinh[(1 + a Cos[2x])/(a^2Sin[2x]^2 + b^2)^(1/2)] + ArcSinh[(a + Cos[2x])/(Sin[2x]^2 + b^2)^(1/2)]);
T6 = -2b Sin[2x](a^2ArcTan[b (a + a Cos[2x])/(a Sin[2x]r2)] + ArcTan[b (1 + Cos[2x])/( Sin[2x]r1)]);
T7 = 2b Sin[2x](a^2ArcTan[b (1 + a Cos[2x])/(a Sin[2x]r)] + ArcTan[b (a + Cos[2x])/( Sin[2x]r)]);
T8 = Sin[2x]^2/3(a^3Log[(r + 1 + a Cos[2x])/(r-1-a Cos[2x])] + Log[(r + a + Cos[2x])/(r-a- Cos[2x])]);
T9 = -Sin[2x]^2/3(a^3Log[(r0 + 1 + a Cos[2x])/(r0-1-a Cos[2x])] + Log[(r0 + a + Cos[2x])/(r0-a- Cos[2x])]);
T10 = -Sin[2x]^2/3(a^3Log[(r2 + a + a Cos[2x])/(r2-a-a Cos[2x])] + Log[(r1 + 1 + Cos[2x])/(r1-1- Cos[2x])]);
T11 = 2/3(a^3 + 1) Sin[2x]^2Log[Cos[x/2]/Sin[x/2]];
f = Cos[2x](T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 + T9 + T10 + T11);
A = NIntegrate[f,{x,0,Pi/2},WorkingPrecision->30, AccuracyGoal->30];
N[A,16];
B = -2mu n1^2 R1/(b^2Log[a]^2);
N[B,16];
L = A B;
N[L,16]
0.0004383988542717143
L = 0.0004383988542717143 (H) = 0.4383988542717143 (mH)
Appendix B
MATHEMATICA CODE (AZIMUTHAL CURRENT)-EXAMPLE 4 (case .
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mu = 4Pi/10000000;
n1 = 1;
R1 = 1;
R2 = 3;
l = 2;
a = R2/R1;
b = l/R1;
n1 = 1;
r = (a^2 + 2 a Cos[2x] + 1 + b^2)^(1/2);
r0 = (a^2 + 2 a Cos[2x] + 1)^(1/2);
r2 = (2a^2 + 2 a^2Cos[2x] + b^2)^(1/2);
r02 = (2a^2 + 2 a^2Cos[2x])^(1/2);
r1 = (2 + 2 Cos[2x] + b^2)^(1/2);
r01 = (2 + 2 Cos[2x])^(1/2);
f11 = b^4/Sin[2x]^2( r2-b Cos[2x]/Sin[2x] ArcTan[(a ^2Sin[2x]^2-b^2 Cos[2x])/(b Sin[2x]r2)]);
f12 = b^4/Sin[2x]^2( r1-b Cos[2x]/Sin[2x] ArcTan[(Sin[2x]^2-b^2 Cos[2x])/(b Sin[2x]r1)]);
f13 = -2b^4/Sin[2x]^2( r-b Cos[2x]/Sin[2x] ArcTan[(a Sin[2x]^2-b^2 Cos[2x])/(b Sin[2x]r)]);
S1 = f11 + f12 + f13;
f21 = 9a^2b^2r2 + 2(6a^4Cos[2x]^2-2a^4Cos[2x]-8a^4)(r2-r02);
f22 = 9b^2r1 + 2(6Cos[2x]^2-2Cos[2x]-8)(r1-r01);
f23 = -9b^2(a^2 + 1)r -4(3(a^4 + 1)Cos[2x]^2-a(a^2 + 1)Cos[2x]-2(a^2 + 1)^2)(r-r0);
S2 = f21 + f22 + f23;
f31 = 30b Sin[2x]Cos[2x](a^4ArcTan[b (1 + Cos[2x])/( Sin[2x]r2)] + ArcTan[b (1 + Cos[2x])/( Sin[2x]r1)]);
f32 = -30b Sin[2x]Cos[2x](a^4ArcTan[b (1 + a Cos[2x])/( a Sin[2x]r)] + ArcTan[b (a + Cos[2x])/( Sin[2x]r)]);
S3 = f31 + f32;
S4 = 15b (a^4Sin[2x]^2Log[(r2 + b)/(r2-b)] + Sin[2x]^2Log[(r1 + b)/(r1-b)]-((a^2 + 1)^2-2(a^4 + 1)Cos[2x]^2)/2Log[(r + b)/(r-b)]);
S5 = 12Cos[2x]Sin[2x]^2(a^5Log[r0 + 1 + a Cos[2x]] + Log[r0 + a + Cos[2x]]-(a^5 + 1)Log[4Cos[x]Cos[x/2]^2]-a^5Log[a]);
f61 = 4Cos[2x](5b^2-3Sin[2x]^2)Log[(r + a + Cos[2x])/(r1 + 1 + Cos[2x])];
f62 = 4a^3Cos[2x](5b^2-3a^2Sin[2x]^2)Log[(r + 1 + a Cos[2x])/(r2 + a + a Cos[2x])];
S6 = f61 + f62;
f = Cos[2x](S1 + S2 + S3 + S4 + S5 + S6);
A = NIntegrate[f,{x,0,Pi/2},WorkingPrecision->30, AccuracyGoal->30];
N[A,16];
B = -mu n1^2 R1/15/b^2/(a-1)^2;
N[B,16];
L = A B;
N[L,16]
2.533006546891938*10−6
L = 2.533006546891938 (μH)