r[x_, y_] := Sqrt[x^2 + y^2];
R2[x_, y_] := r[x, y]^2 + 1 + 2/r[x, y];
phidot[x_, y_] := If[r[x, y] >= 2, 2/((r[x, y])^3 + r[x, y] + 2), 0];
vx[x_, y_] := -y phidot[x, y];
vy[x_, y_] := x phidot[x, y];
VectorPlot[{vx[x, y], vy[x, y]}, {x, -2 [Pi], 2 [Pi]}, {y, -2 [Pi],
2 [Pi]}]