Vector Potential of a Continuous Ring

In this worksheet you will examine the vector potential due to a continuous ring of charge with radius R=1, linear charge density given by ((λ (Subscript[μ, o]))/(4π))=1, and rotating with period T=1.

The vector potential is given by an elliptic integral, which Mathematica cannot do in general.

(OverVector[A])=(∫_0^(2π))((OverHat[ϕ'])/(Sqrt[(r^2)+1-2r cos(ϕ-ϕ')+(z^2)]))dϕ'   

Since we can not integrate (OverHat[ϕ']) as it stands, we rewrite it using rectangular basis vectors as (OverHat[ϕ'])=-sin(ϕ')(OverHat[x])+cos(ϕ')(OverHat[y]).  We take advantage of axial symmetry to set ϕ=0.

Clear[r,z,ϕ]

(Subscript[A, x])=Integrate[-Sin[ϕ']/Sqrt[r^2+1-2r Cos[ϕ']+z^2],{ϕ',0,2π},Assumptions->{r>0,z>0}]

(Subscript[A, y])=Integrate[Cos[ϕ']/Sqrt[r^2+1-2r Cos[ϕ']+z^2],{ϕ',0,2π},Assumptions->{r>0,z>0}]

?EllipticK

?EllipticE

Mathematica can "do" the integral for particular values of the coordinates r, ϕ, and z.

(Subscript[A, y])/.{r->4,z->0}

N[%]

It can also use this numerical evaluation to plot the potential We plot (Subscript[A, y]) as a color on the y-z plane

Plot3D[(Subscript[A, y]),{r,0,3},{z,-1.5,1.5},AxesLabel->{r,z}, ColorFunction-> Hue,ViewPoint->{0,0,Infinity},MeshFunctions->{#3 &}]

or we can plot horizontal cross-sections (constant values of z) for the potential.

This time we compute (Subscript[A, y])=(∫_0^(2π))((cos(ϕ'))/(Sqrt[(r^2)+1-2r cos(ϕ')+(0.1^2)]))dϕ'

<<Graphics`ParametricPlot3D`

CylindricalPlot3D[(Subscript[A, y])/.z->0.1,{r,0,1.2},{ϕ,0,2π},AxesLabel->{"x","y","z"},ColorFunction-> Hue,ViewPoint->{0,0,Infinity},MeshFunctions->{#3 &}]

by Corinne Manogue