Electric Field with line charge

Electric Field at point [0,d]

In[]:=
_0[ε_,d_,x_l_,x_r_]=Assuming[ε>0&&d>0&&x_l<0&&x_r>0,{Integrate[-x/(x^2+d^2)^(3/2),{x,x_l,x_r}],Integrate[d/(x^2+d^2)^(3/2),{x,x_l,x_r}]}/(2Pi*ε)]
Out[]=

-
1
2
d
+
2
x_l
+
1
2
d
+
2
x_r
2πε
,
-
x_l
2
d
+
2
x_l
+
x_r
2
d
+
2
x_r
2dπε

In[]:=
E_0_x[ε_,d_,x_l_,x_r_]=_0[ε,d,x_l,x_r][[1]]
Out[]=
-
1
2
d
+
2
x_l
+
1
2
d
+
2
x_r
2πε
In[]:=
E_0_y[ε_,d_,x_l_,x_r_]=_0[ε,d,x_l,x_r][[2]]
Out[]=
-
x_l
2
d
+
2
x_l
+
x_r
2
d
+
2
x_r
2dπε

E_x/E_y under x_l=-10, x_r = α*10

In[]:=
Plot[E_0_x[8.85/10^12,1,-10,α*10]/E_0_y[8.85/10^12,1,-10,α*10],{α,0.5,20},PlotRange->Full,AxesLabel->{"α","E_0_x/E_0_y"}]
Out[]=
In[]:=
[x_,y_]=_0[8.85/10^12,y,-20-x,20-x]
Out[]=
1.79836×
10
10
-
1
2
(-20-x)
+
2
y
+
1
2
(20-x)
+
2
y
,
1.79836×
10
10
-
-20-x
2
(-20-x)
+
2
y
+
20-x
2
(20-x)
+
2
y
y


Visualized electric field when x_l=-20, x_r=20

In[]:=
VectorPlot[[x,y],{x,-40,40},{y,-40,40},RegionFunction->Function[{x,y},Abs[y]>0.01],VectorPoints->Fine,AspectRatio->Automatic]
Out[]=