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Electric field Plot

Approximate the potential and the electric field, on the axis of a possitively charged ring

H.Tahsiri

Off[General::spell];
Off[General::spell1];
Pagewidth->70;

Clear[Vp,Q,R,k,y]

1) Find the Electric potential

Vp[y_]=k Q/Sqrt[y^2+R^2]

          k Q
     -------------
           2    2
     Sqrt[R  + y ]

2) Taylor expand the potential, up to a second order, near the center of the ring

Vpnear=Series[Vp[y],{y,0,2}]//Normal//Simplify

             2    2
     k Q (2 R  - y )
     ---------------
            2 3/2
        2 (R )

3) Find the potential far away from the center of the ring

Vpfar=Series[Vp[y],{y,Infinity,2}]//Normal

     k Q
     ---
      y

4) Find the electric field , by taking the first derivative or the gradient of the potential found in (1)

Ep=-D[Vp[y],{y,1}]

        k Q y
     ------------
       2    2 3/2
     (R  + y )

5) Taylor expand the electric field up to a second order

Epnear=Series[Ep,{y,0,2}]//Normal

      k Q y
     -------
       2 3/2
     (R )

6) Find the electric field far from the center of the ring ( a point charge approximations)

Epfar=Series[Ep,{y,Infinity,3}]//Normal

     k Q
     ---
      2
     y

7) Plot of the electric field as a function of y ( in blue )

given={k->1,Q->1,R->1};
Epplot=Plot[Ep/.given,{y,-5,5},PlotRange->{{-5,5},{-.5,.5}},
PlotStyle->{Hue[0.6]},AxesLabel->{"y",""},PlotLabel -> 
      StyleForm[TraditionalForm["E=(k Q y)/(R^2 + y^2)^(3/2)"], 
        FontSize -> 12]];

8) Plot of the electric field as a function of y near y=0 (near field approximation )

Epnearplot=Plot[Epnear/.given,{y,-5,5},PlotRange->{{-5,5},{-.5,.5}},
PlotStyle->{Hue[0.9]},AxesLabel->{"y",""},PlotLabel -> 
      StyleForm[TraditionalForm["E=(k Q y)/ R^3 ( E near center )"], 
        FontSize -> 12]];

9) Plot of the electric field as a function of y, far from the center ( far field approximations)

Epfarplot=Plot[Epfar/.given,{y,-5,5},PlotRange->{{-5,5},{-100,100}},
PlotStyle->{Hue[.4]},AxesOrigin->{0,0},
AxesLabel->{"y",""},AxesLabel->{"E","y"},PlotLabel -> 
      StyleForm[TraditionalForm["E=(k Q)/ y^2,(E far from center)"], 
        FontSize -> 12]];

Epsurfacetopbottom3=SurfaceOfRevolution[Epfar/.given,{y,-8,8},
PlotRange->All,RevolutionAxis->{1,0,0},
Boxed->False,AxesEdge->None,
ViewVertical->{0.,1.,1.},ImageSize->250,PlotLabel -> 
StyleForm[TraditionalForm["E=(k Q)/ y^2,( y verses E )"], 
FontSize -> 12]];

10 ) Show all of the plots together

text=Graphics[{Text["NearField(red)",{4.3,.2}],
Text["FarField(green)",{4.3,.3}],Text["E-Field(blue)",{4.1,.4}]}];

Eplotall=Show[{Epplot,Epnearplot,Epfarplot},PlotLabel-> Electric Field Plots,
PlotRange->{{-5,5},{-.5,.5}},AxesLabel->{"y",""},Epilog->text[[1]],
PlotLabel->"E-Fields"];

1) E-field plot verses y. (no approximation)

2) E-field plot for a distance near the center of the ring.( a near point approximation )

1) E-field plot for a distace far from the center of the ring.(a point charge approximation)

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