Conformal Mapping

90 degree corner; f=z^2; z=f^(1/2)

x[u_,v_]=((Sqrt[u+(Sqrt[(u^2)+(v^2)])])/(Sqrt[2]));
y[u_,v_]=(((-u+(Sqrt[(u^2)+(v^2)])) (Sqrt[u+(Sqrt[(u^2)+(v^2)])]))/((Sqrt[2]) v));

z[u_,v_]={x[u,v],y[u,v]};

g[u_]=Table[z[u,v],{v,0.0000001,2,.2}];

Equipotentials=ParametricPlot[Evaluate[g[u]],{u,-2,2},AspectRatio->Automatic,PlotRange->All,PlotStyle->RGBColor[1,0,0]]

h[v_]=Table[z[u,v],{u,-1,1,.2}];

ElectricField=ParametricPlot[Evaluate[h[v]],{v,0.0001,2},AspectRatio->Automatic,PlotRange->All,PlotStyle->RGBColor[0,1,0]]

Show[ElectricField,Equipotentials]