Electric Lines of Force of Infinitesimal Dipole
微小ダイポールから放射される電磁界の電気力線

By Takuichi Hirano (Tokyo Institute of Technology)

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Infinitesimal Dipole
微小ダイポール

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η = 120 * π ;   (* 界インピーダンス *)

k = 2 * π ;   (* 位相定数 *)

Er[r_, θ_] := η (k^2 I ℓ E^(-I k r))/(2 π) (1/(k r)^2 - I/(k r)^3) Cos[θ] ;

Eθ[r_, θ_] := η (k^2 I ℓ E^(-I k r))/(4 π) (I/(k r) + 1/(k r)^2 - I/(k r)^3) Sin[θ] ;

Hφ[r_, θ_] := η (k^2 I ℓ E^(-I k r))/(4 π) (I/(k r) + 1/(k r)^2) Sin[θ] ;

\[DoubleStruckCapitalE] = Overscript[r, ∧] E_r + Overscript[θ, ∧] E_θ

Prepare Animation
アニメーションの準備

R = k r

T = ω t

K = η   (k^2 I ℓ)/(4 π)

By the variable transformation above

Er E^(I T) = 2 K E^(I (T - R)) (1/R^2 - I/R^3) Cos[θ]

Eθ  E^(I T) = K E^(I (T - R)) (1/R + I/R^2 - I/R^3) Sin[θ]

Peak expression of the electric field is written as

e_R = 2 K Cos[θ]    (Cos[T - R]/R^2 + Sin[T - R]/R^3)

e_θ = K Sin[θ] [Cos[T - R]/R^2 - (1/R - 1/R^3) Sin[T - R]]

Differential eq . for the electric field is

dR/e_R = (R dθ)/e_θ

((1 - 1/R^2) Sin[T - R] + Cos[T - R]/R)/(Cos[T - R] + Sin[T - R]/R) dR = 2 Cos[θ]/Sin[θ] dθ         (Variables separable)

Then, integrate this equation . <br /> -Log[Cos[T - R] + Sin[T - R]/R] = 2 Log[ Sin[θ] ] + C '

Finnary, <br />[Cos[R - T] + Sin[R - T]/R]    (Sin[θ])^2 = const .

Animation of Electric Lines of Force
(ContourPlot)

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l = 0.2 ;

dip = {RGBColor[1, 0, 0], Arrow[{0, -l/2}, {0, l/2}]} ;

ncell = 8 ;

k0 = 2 * π ;

R[x_, y_] := k0 (x^2 + y^2)^(1/2) ;

θ[x_, y_] := ArcTan[x/y] ;

contrs = Table[c, {c, -0.9, 0.9, 0.2}] ;

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