Some integrals that didn’t evaluate in reasonable time numerically.  Numeric integration of the same, and a Manipulate to show the integrand.  Also a plot (chargeAndCurrentOnRingFig1.eps) that shows the geometry of the ring configuration and the coordinates used.

$Assumptions = z ∈ Reals && r ∈ Reals && r ≠ 1;

(*Integrate[ (z^2 + 1 + r^2  - 2 r  Cos[phi])^(-3/2), {phi, 0, 2 Pi}]
Integrate[ Cos[ phi] (z^2 + 1 + r^2  - 2 r  Cos[phi])^(-3/2), {phi, 0, 2 Pi}]*)
(*Integrate[ Sin[ phi] (z^2 + 1 + r^2  - 2 r  Cos[phi])^(-3/2), {phi, 0, 2 Pi}]*)
Integrate[ Sin[ phi] ( 1 + r^2  - 2 r  Cos[phi])^(-3/2), {phi, 0, 2 Pi}]
(*Integrate[ E[I phi] (z^2 + 1 + r^2  - 2 r  Cos[phi])^(-3/2), {phi, 0, 2 Pi}]*)

ClearAll[a,b,c]
a[u_, v_, p_:8] := NIntegrate[ (u^2 + 1 + v^2  - 2 v  Cos[phi])^(-3/2), {phi, 0, 2 Pi}, AccuracyGoal->p]
b[u_, v_, p_:8] := NIntegrate[ Cos[phi] (u^2 + 1 + v^2  - 2 v  Cos[phi])^(-3/2), {phi, 0, 2 Pi}, 
AccuracyGoal->p]
c[u_, v_, p_:8] := NIntegrate[ Sin[phi] (u^2 + 1 + v^2  - 2 v  Cos[phi])^(-3/2), {phi, 0, 2 Pi}, AccuracyGoal->p]

Manipulate[

(*
{
(*z,
r,*)
a[z,r]
,b[z,r]
, c[z,r]
} // Column*)
Plot[ Sin[p] (z^2 + 1 + r^2  - 2 r  Cos[p])^(-3/2), {p,0,2 Pi}, PlotRange -> Full,
Ticks->{{0, Pi/2,Pi,3Pi/2, 2Pi}, {-1,1}}]
,
{z,0,10},
{r,0,0.9}(*,*)
(*{r,1.1,2}*)
]

ClearAll[bold,fs,te]
bold = Style[ #, Bold] &;
fs:= Style[ #, FontSize -> 16 ]&;
te[n_] := Subscript["e"// bold, n]//fs;

p1=Module[{p,r, e1, e2, e3, o,x,xp, rad, phip1,phip2, es, phicap, rhocap},
p = 40;
rad = 1.5;
r = {-1,1}rad;
o = {0,0,0};
es = 0.7;
{e1,e2,e3} =  es IdentityMatrix[3];
x =  e1 +0.5e2+ 1.5 e3;
phip1 = -0.33Pi;
phip2 = 1.1 Pi;
xp = (e1 Cos[phip1] + e2 Sin[phip1])/es;
phicap[phi_] := (e2 Cos[phi] - e1 Sin[phi])/es;
rhocap[phi_] := (e1 Cos[phi] + e2 Sin[phi])/es;
Show[
{
RevolutionPlot3D[{(p+Cos[t])/p,Sin[t]/p},{t,0,2Pi}, PlotRange->{r,r,{-0.1,rad}}, Ticks->None
,Mesh->1],
Graphics3D[ 
{
Black,
Arrow[Tube[{o,e1}]],
Arrow[Tube[{o,e2}]],
Arrow[Tube[{o,e3}]],
Red//Darker,
Arrow[Tube[{o,x}]],
Green//Darker,
Arrow[Tube[{o,xp}]],
Blue//Darker,
Arrow[Tube[{rhocap[phip2],rhocap[phip2]+ phicap[phip2]}]],

Black,
Text[te[1], 1.2e1],
Text[te[2], 1.2e2],
Text[te[3], 1.2e3],
Text[Superscript[OverHat["ρ"]// fs//bold, "'"], 1.2 xp],
Text["x" // fs//bold,1.1 x],
Text["v" // fs//bold, 1.1rhocap[phip2]+ 0.5phicap[phip2]],
Blue//Darker,
Text["R"//fs,0.5rhocap[1.1 Pi]],
Thick,
Line[{o,rhocap[1. Pi]}]
}
]
}
,ViewPoint->{2.3,-2.4`,2.`}
]
]

<<peeters` ;
peeters`setGitDir[ "../project/figures/GAelectrodynamics" ]

peeters`exportForLatex["chargeAndCurrentOnRingFig1", p1]