θnew[r0_] := ArcTan[r Sin[θ]/(r Sin[θ] - r0)]

field =
  {FullSimplify[Cos[num (θ + α)] Cos[θ] + Sin[num (θ + α)] Sin[θ]],
   FullSimplify[-Cos[num (θ + α)] Sin[θ] + Sin[num (θ + α)] Cos[θ]]} /.
  num -> 1 /. α -> Pi/2 /. θ -> θnew[0.5]

VectorPlot[
  Evaluate @ TransformedField["Polar" -> "Cartesian", field, {r, θ} -> {x, y}],
  {x, -1, 1}, {y, -1, 1},
  RegionFunction -> Function[{x, y}, x^2 + y^2 > .2]]

Manipulate[
 StreamPlot[
  Evaluate[
   s/3 ( Grad[1/Sqrt[(x + s)^2 + y^2], {x, y}] -
       Grad[1/Sqrt[(x - s)^2 + y^2], {x, y}]) + {0.2, 0}],
  {x, -4, 4}, {y, -4, 4}, StreamMarkers -> "Pointer",
  StreamPoints -> Fine],
 {s, 0, 2}
 ]