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  <item rdf:about="https://community.wolfram.com/groups/-/m/t/3645153">
    <title>Currents in a closed dish</title>
    <link>https://community.wolfram.com/groups/-/m/t/3645153</link>
    <description>Hello all!&#xD;
&#xD;
I am trying to solve a problem of calculating induced electrical currents in a varying magnetic field. The field originates from current through a coil. I am only interested in the steady state solution, but also the time dependence might be interesting to solve.&#xD;
&#xD;
I already calculated the A field from the coils. There are two coils, each has 2 layers with 5 turns each, so a total of 2 turns (see code below). This is the field for a constant current of 1A, but my actual field would have a slope of e.g. 1 A/s.&#xD;
&#xD;
What I am interested now is:&#xD;
&#xD;
A circular dish is filled with conducting medium, and no current across the dish walls.The dish has a size of e.g. 35mm diameter and 5 mm height, and its bottom center is at the origin. How do the currents look that are induced by the changing field? Can we also determine the distribution of charges?&#xD;
&#xD;
    (*Magnetic Field Simulation for Two Coils*)&#xD;
    &#xD;
    (*Coil Parameters*)&#xD;
    x0 = 0.04;           (* Coil axis offset from origin = 40 mm *)&#xD;
    n0 = 5;                  (* Number of turns in each layer *)&#xD;
    d1 = 0.005;            (* distance of turns = 5 mm *)&#xD;
    r0 = x0 - (n0 - 1/2)*d1;         (* Inner coil radius *)&#xD;
    &#xD;
    m0 = 2;                  (* Number of layers *)&#xD;
    d0 = 0.014;         (* distance of layers = 14 mm *)&#xD;
    d2 = 0.006;        (* offset from origin = 6 mm *)&#xD;
    i0 = 1.0;              (* Current = 1A *)&#xD;
    &#xD;
    (* Permeability of free space[H/m] *)&#xD;
    mu0 = 4*Pi*10^-7; &#xD;
    &#xD;
    (*Each coil is defined by {xc, r, zc, I}*)&#xD;
    leftCoilTurns = &#xD;
      Flatten[Table[&#xD;
        Table[{-x0, r0 + n*d1, -l*d0 - d2, i0}, {n, 0, n0 - 1}], {l, 0, &#xD;
         m0 - 1}], 1];  &#xD;
    rightCoilTurns = &#xD;
      Flatten[Table[&#xD;
        Table[{x0, r0 + n*d1, -l*d0 - d2, -i0}, {n, 0, n0 - 1}], {l, 0, &#xD;
         m0 - 1}], 1];  &#xD;
    allTurns = Join[leftCoilTurns, rightCoilTurns];&#xD;
    &#xD;
    AFieldTurn[{x_, y_, z_}, {xc_, yc_, zc_}, r_, i_] := &#xD;
     Module[{dx, dy, dz, rho, k2, k, aphi},&#xD;
      dx = x - xc; dy = y - yc; dz = z - zc;&#xD;
      rho = Sqrt[dx^2 + dy^2];&#xD;
      If[rho != 0,&#xD;
       rho = Sqrt[dx^2 + dy^2];&#xD;
       k2 = 4 r*rho/((r + rho)^2 + dz^2);&#xD;
       k = Sqrt[k2];&#xD;
       aphi = mu0*i/(Pi*k)*Sqrt[r/rho] ((1 - k2/2) EllipticK[k2] - EllipticE[k2]);&#xD;
       {-aphi*dy/rho, aphi*dx/rho, 0},&#xD;
       {0, 0, 0}]&#xD;
      ]&#xD;
    &#xD;
    AFieldTotalXYZ[x_?NumericQ, y_?NumericQ, z_?NumericQ] := &#xD;
     Plus @@ (AFieldTurn[{x, y, z}, {#[[1]], 0, #[[3]]}, #[[2]], #[[&#xD;
           4]]] &amp;amp; /@ allTurns)&#xD;
    AMagnitude[x_, y_, z_] := Norm[AFieldTotalXYZ[x, y, z]]&#xD;
    ContourPlot[AMagnitude[x, 0, z], {x, -0.1, 0.1}, {z, 0, 0.05}, &#xD;
     Contours -&amp;gt; 20, ColorFunction -&amp;gt; &amp;#034;Rainbow&amp;#034;, AspectRatio -&amp;gt; Automatic,&#xD;
      PlotPoints -&amp;gt; 30]&#xD;
    ContourPlot[AMagnitude[x, y, 0], {x, -0.1, 0.1}, {y, -0.05, 0.05}, &#xD;
     Contours -&amp;gt; 20, ColorFunction -&amp;gt; &amp;#034;Rainbow&amp;#034;, AspectRatio -&amp;gt; Automatic,&#xD;
      PlotPoints -&amp;gt; 30]&#xD;
&#xD;
&#xD;
Thank you for your ideas and help!&#xD;
&#xD;
Max</description>
    <dc:creator>Maximilian Ulbrich</dc:creator>
    <dc:date>2026-02-25T15:24:32Z</dc:date>
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