Element[{R, x, z},Reals]

Integrate[(x-R Cos[p])/Sqrt[x^2+z^2+R^2-2 x R Cos[p]]^3,{p,-Pi,Pi}, Assumptions -> x > 0 && R > 0 && z > 0]
ex=%

Integrate[(z)/Sqrt[x^2+z^2+R^2-2 x R Cos[p]]^3,{p,-Pi,Pi}, Assumptions -> x > 0 && R > 0 && z > 0]
ez=%

{ex,ez} /. R->1
VectorPlot[%,{x,-2,2},{z,-2,2},VectorPoints->16, RegionFunction -> Function[{x, z, vx, vy, n}, (x-1)^2+z^2>.10 && (x+1)^2+z^2>.10],VectorScale->.25,PlotRangePadding ->None,PlotRange->2  ]

VectorPlot[%%, {x, -2, 2}, {z, -2, 2},VectorPoints->16, RegionFunction -> Function[{x, z, vx, vy, n}, (x-1)^2+z^2>.10],VectorScale->{Small,Automatic,None} ]

ez /. x->0
           2 Pi z
Out[8]= ------------
          2    2 3/2
        (R  + z )

Integrate[1/Sqrt[x^2+z^2+R^2-2 x R Cos[p]],{p,-Pi,Pi}, Assumptions -> x > 0 && R > 0 && z > 0]
phi=% /. R->1
ContourPlot[phi, {x, -2, 2}, {z, -2, 2},ContourShading ->False,Contours -> 16,PlotRangePadding ->None]

v={-D[phi,x],-D[phi,z]}
VectorPlot[v,{x,-2,2},{z,-2,2},VectorPoints->16, RegionFunction -> Function[{x, z, vx, vy, n}, (x-1)^2+z^2>.10 && (x+1)^2+z^2>.10],VectorScale->.25,PlotRangePadding ->None,PlotRange->2  ]

Show[%,%%%]
