f[c_,n_]=2(-Integrate[LegendreP[n,x],{x,0,c}]+Integrate[LegendreP[n,x],{x,c,1}])
FindRoot[f[c,4],{c,.6}]
g[n_]=f[c,n]/.%
a=Table[N[g[n] (2 n+1)/2],{n,0,10,2}]
phi[r_,c_]=Sum[a[[i]] r^(2 i-2) LegendreP[2 i-2,c],{i,1,6}]

phi2[r_,c_]=Sum[a[[i]] r^(2 i-2) LegendreP[2 i-2,c],{i,1,2}]

ContourPlot[phi[Sqrt[x^2+z^2],z/Sqrt[x^2+z^2]],{x,-.9,.9},{z,-.9,.9},ContourShading->False,RegionFunction->Function[{x,z},(x)^2+z^2<.9^2],Contours->Table[.05*i+phi[0,0],{i,-10,22}],PlotRangePadding->None]

Plot[{phi[r,1],phi[r,0],phi2[r,1],phi2[r,0]},{r,0,.9},PlotStyle->{Directive[Black],Directive[Black],Directive[Red],Directive[Red]}]

#hi