slab geometry
E-field and current are proportional

j[x_] := (E^((I - 1) x/δ))

Plot[Re[j[x δ]], {x, 0, 6}, 
        PlotRange -> All];

Plot[Abs[j[x δ]], {x, 0, 6}, 
        PlotRange -> All];

j[x_] := (E^((I - 1) (x - w)/δ)) +  (E^(-(I - 1) (x + w)/δ))

change value of the skin depth δTst, to vary penetration of field into the conductor

δTst = 1;
Plot[
        Abs[j[x]/j[w]] /. {w -> 
              1, δ -> δTst}, {x, -1, 1}, 
        PlotRange -> {{-1, 1}, {0, 1}}];

cylindrical/wire geometry: see HM 5.6 for derivation and notation

2 π Integrate[
        BesselJ[0, (Sqrt[2 I]) r/δ] r, {r, 0, 
          a}]

Plot[
        Re[BesselJ[0, (Sqrt[2 I]) u]]/Re[BesselJ[0, (Sqrt[2 I]) 1]], {u, 0, 1}, PlotRange -> All];

Plot[Abs[BesselJ[0, (Sqrt[2 I]) u]], {u, 0, 1}, 
        PlotRange -> All];

uMax = 7.55;
Plot[
        Abs[BesselJ[0, (Sqrt[2 I]) uNorm uMax]/BesselJ[0, (Sqrt[2 I]) uMax]], {uNorm, 0, 1}, 
        PlotRange -> All];