Dielectric Sphere in an External Field

Clear["`*"]

χ  =  20

E0  =  2

Ei  =  3/(3 + χ) E0

a = 1

b =  (χ/(3 + χ)) a^3 E0

Φ  =  
      Piecewise[{{-E0 z  + b z/r^3, r > a}, {-Ei z, 
            r < a}}]

ϕ[x_, z_]  =  Φ  /.  
        r ->  Sqrt[x^2 + z^2]

Needs["Graphics`PlotField`"]

cd  =  
      ContourPlot[ϕ[x, z], {x, -2, 2}, {z, -2, 
          2}, 
PlotPoints -> 200, 
        ColorFunction -> 
          Hue, 
DisplayFunction ->  
          Identity]

ce  =  
      PlotGradientField[-ϕ[x, z], {x, -2, 2}, {z, -2, 
          2}, 
ScaleFunction ->  (1 &),  
        DisplayFunction ->  Identity]

cir = Graphics[{Thickness[0.008], Circle[{0, 0}, 1]}]

Show[{cd, ce, cir}, 
      DisplayFunction ->  $DisplayFunction]

Method of Images

Consider a charge q imbedded in a dielectric (Subscript[ϵ, 1]) at a distance d above a plane interface with a second dielectric (Subscript[ϵ, 2]). Calculate the potential:

Clear["`*"]

ϕ1  :=  
      1/(4  π ϵ1) (q/Sqrt[(z - d)^2 + ρ^2] + 
            qi/Sqrt[(z + d)^2 + ρ^2])

ϕ2  :=  
      1/(4  π ϵ2) 
        qj/Sqrt[(z - d)^2 + ρ^2]

D1  :=  -ϵ1 D[ϕ1, z]

D2  :=  -ϵ2 D[ϕ2, z]

c1  :=  D1 - D2 /.  z ->  0

c2  :=  Simplify[ϕ1 - ϕ2] /.  z -> 0

sol  =  
      Solve[{c1 == 0, c2 == 0}, {qi, qj}] // First

qi  =  qi /.  sol

qj  =  qj /.  sol

Choose values for q and d

q = 1

d = 1

Let's look at case of transition from vacuum to polarized medium

ϵ1  =  1

ϵ2  =  4

ϕ[ρ_, z_]  =  
      Piecewise[{{ϕ1, z > 0}, {ϕ2, z < 0}}]

Plot3D[ϕ[x, z], {x, -2, 2}, {z, -2, 2}]

Needs["Graphics`PlotField`"]

cd  =  
      ContourPlot[ϕ[x, z], {x, -2, 2}, {z, -2, 
          2}, 
PlotPoints -> 200, 
        ColorFunction -> Hue,  
        DisplayFunction ->  Identity]

ce  =  
      PlotGradientField[-ϕ[x, z], {x, -2, 2}, {z, -2, 
          2}, 
ScaleFunction ->  (1 &),  
        DisplayFunction ->  Identity]

Show[{cd, ce}, DisplayFunction ->  $DisplayFunction]

Bound charge at interface  (should be negative for (Subscript[ϵ, 2])>(Subscript[ϵ, 1])), positive otherwise.

σb  =  (D1/ϵ1 - D2/ϵ2) /.  
        z ->  0

Now go back and consider transition from polarized material to vacuum