POLARIZATION OF DIELECTRIC SPHERE IN DIELECTRIC MATRIX IN PRESENCE OF ELECTROSTATIC FIELD El Hassan EL HAROUNY (a), El Mahdi ASSAID* (b) (c), Asmaa IBRAL (b) (c), Soukaina NAKRA MOHAJER (a) and Jamal EL KHAMKHAMI (a) (a) Laboratoire de Physique de la Matière Condensée, Département de Physique, II, 93030 Tétouan, Royaume du Maroc. (b) Equipe d'Electronique et Optique des Nanostructures & (c) Laboratoire d'Instrumentation, Mesure et Contrôle, Département de Physique, Faculté des Sciences, Université Chouaib Doukkali, El Jadida, Royaume du Maroc.
[email protected], © 2017 El Mahdi Assaid * Corresponding author
ABSTRACT In this worksheet, we investigate the polarization of a dielectric sphere (dot) with a relative permittivitty embedded in a dielectric matrix with a relative permittivitty
and submitted to an uniform electrostatic field
F oriented in z-axis direction. It's a fondamental and popular problem
present in most of electromagnetism textbooks. First of all, we express Poisson equation in appropriate coordinates system: V(r, , ) = 0. We proceed to a full separation of variables and derive general expression of scalar electrostatic potential V(r, , ). Then we particularize to a dielectric sphere surrounded by a dielectric matrix and give expressions of electrostatic potential V(r, ) in the meridian plane (x0z) inside and outside the sphere by taking into account: i) invariance property of the system under rotation around z-axis, ii) choice of the plane z=0 as a reference of scalar electrostatic potential, iii) regularity of V(r, ) at the origine and very far from the sphere, iv) continuity condition of scalar electrostatic potential V(r, ) at the sphere surface, v) continuity condition of normal components of electric displacement field D at the sphere surface. The obtained expressions of V(r, ) inside and outside the sphere allows as to derive expressions of electrostatic field F, electric displacement field D and polarization field P inside and outside dielectric dot in spherical coordinates as well as in cartesian rectangular coordinates. The paper is a proof of Maple algebraic and graphical capabilities in tackling the resolution of Poisson equation as a second order partial differential equation and also in displaying scalar electrostatic potential contourplot, electrostatic field lines as well as fieldplots of F, D and P inside and outside dielectric sphere.
SUBJECTS Electromagnetism; Nanophysics.
SCHEMATIC REPRESENTATION
The figure above displays contourplot of scalar electrostatic potential V(r, ) (green lines), electrostatic field F lines (red lines) and F fieldplot (red arrows) in meridian plane (x0z) of silicon dielectric sphere with a radius equal to 10 cm in vacuum.
INTRODUCTION In spherical coordinates and in the absence of any electrostatic charge, Poisson equation reduces to Laplace equation [1]: V(r, , ) = 0. V(r, , ) is scalar electrostatic potential. In the present case, Poisson equation is a fully separated second order homogeneous partial differential equation. So, we may express scalar electrostatic potential as a product of
three functions: V(r, , ) = R(r) ( ) ( ). In this way, we show using Maple algebraic capabilities that:
In the meridian plane (x0z) , we have:
In the following, we use the symmetry of the system and the boundary conditions to find expressions of potential which are acceptable from physical point of view. The invariance property of the system under rotation around z-axis impose m=0 . So:
The choice of potential reference at z=0 impose l=0. So, the general expression of potential we are looking for is:
In this way, the expressions of scalar electrostatic potential inside and outside the sphere are:
and
The boundary condition at r=0 implies B to infinity implies A
= 0. The boundary condition at r
= -F.
The continuity condition of electrostatic potential gives [2]: =
The continuity condition of normal component of electric displacement at the sphere surface gives [2]: .n = .n These two latter equations allow us to determine Expressions of scalar electrostatic potential are then used to determine expressions of electrostatic field F=-grad(V), electric displacement D= F and polarization P=D- F inside and outside the dielectric sphere (dot).
INITIALIZATION > restart: with(VectorCalculus): with(LinearAlgebra): SetCoordinates('spherical'[r, theta, phi]): with(PDEtools): with (ScientificConstants): with(plottools): with(plots):
GENERAL SOLUTION OF POISSON EQUATION In this section, we write Poisson equation in spherical coordinates and determine the general expression of scalar electrostatic potential in the meridian plane (x0z). > PDE := expand(Laplacian(V(r, theta, phi)) = 0): pdsolve(PDE, HINT=`*`): _c1 := m^2: _c2 := l*(l+1): ODE[phi] := diff(Phi (phi), phi, phi) = -Phi(phi)*m^2: dsolve(ODE[phi]): ODE[theta] := diff(Theta(theta), theta, theta) = -Theta(theta)*l*(l+1)+ Theta(theta)*m^2/sin(theta)^2-cos(theta)*(diff(Theta(theta), theta))/sin(theta): ODE[x] := simplify(dchange({theta = arccos (x), Theta(theta) = y(x)}, ODE[theta])): dsolve(ODE[x]): ODE[r] := diff(R(r), r, r) = R(r)*l*(l+1)/r^2-2*(diff(R(r), r))/r: dsolve(ODE[r]): V := (m, l, r, theta)->(A*r^l + B/r^(l+1))* LegendreP(l, m, cos(theta));
PARTICULAR SOLUTION IN CASE OF POLARIZED SPHERE In this section, we determine expressions of scalar electrostatic potential inside and outside dielectric sphere submitted to uniform electrostatic field. > m := 0: ODE[phi]: dsolve(ODE[phi]): ODE[x]: dsolve(ODE[x]): ODE [r] := diff(R(r), r, r) = R(r)*l*(l+1)/r^2-2*(diff(R(r), r))/r: dsolve(ODE[r]): V(m, l, r, theta): V1 := : V2 := :
>
V3 := : F_applied := F_0*VectorField(): simplify((VectorNorm(F_applied,2,conjugate=false) )): l := 1: V(m, l, r, theta): E := Gradient(V(m, l, r, theta)): V_Dot := ( r, theta)->(A[Dot]*r + B[Dot]/r^2)*cos(theta): V_Matrix := ( r, theta)->(A[Matrix]*r + B[Matrix]/r^2)*cos (theta): E_Dot := - Gradient(V_Dot( r, theta)): E_Matrix := Gradient(V_Matrix( r, theta)): D_Dot := - epsilon[Dot]*Gradient (V_Dot( r, theta)): D_Matrix := - epsilon[Matrix]*Gradient (V_Matrix( r, theta)): B[Dot] := 0: A[Matrix] := - F_0: equa_1 := simplify(subs([r=R, theta=0],V_Dot( r, theta))=subs([r=R, theta=0],V_Matrix( r, theta))): equa_2 := simplify(subs([r=R, theta=Pi/2],DotProduct(V2, E_Dot))=subs([r=R, theta=Pi/2], DotProduct(V2, E_Matrix))): equa_3 := simplify(subs([r=R, theta= 0],epsilon[Dot]*DotProduct(V1, E_Dot))=subs([r=R, theta=0], epsilon[Matrix]*DotProduct(V1, E_Matrix))): solution := solve( {equa_1, equa_3}, {A[Dot], B[Matrix]}): solution[1]: A[Dot] := rhs(solution[1]): V_Dot( r, theta) := V_Dot( r, theta); B [Matrix] := rhs(solution[2]): V_Matrix( r, theta) := V_Matrix( r, theta);
VECTOR AND SCALAR ELECTROSTATIC CHARACTERISTICS OF DIELECTRIC SPHERE ELECTROSTATIC VECTOR FIELD IN POLAR AND CARTESIAN COORDINATES In this subsection, we determine expressions of electrostatic field inside and outside dielectric sphere in spherical coordinates as well as in cartesian rectangular coordinates. > E_Dot := - Gradient(V_Dot( r, theta)); E_Dot_r := DotProduct (V1, E_Dot); E_Dot_theta := DotProduct(V2, E_Dot); E_Dot_x := simplify(E_Dot_r*sin(theta) + E_Dot_theta*cos(theta)); E_Dot_z := simplify(E_Dot_r*cos(theta) - E_Dot_theta*sin (theta)); E_Matrix := - Gradient(V_Matrix( r, theta)); E_Matrix_r := DotProduct(V1, E_Matrix); E_Matrix_theta := DotProduct(V2, E_Matrix); E_Matrix_x := simplify(E_Matrix_r*
sin(theta) + E_Matrix_theta*cos(theta)); E_Matrix_x := subs( [r=sqrt(x^2+z^2), cos(theta)=z/sqrt(x^2+z^2), sin(theta)= x/sqrt(x^2+z^2)],E_Matrix_x); E_Matrix_z := simplify (E_Matrix_r*cos(theta) - E_Matrix_theta*sin(theta)); E_Matrix_z := subs([r=sqrt(x^2+z^2), cos(theta)=z/sqrt(x^2+ z^2), sin(theta)=x/sqrt(x^2+z^2)],E_Matrix_z); P_0 := F_0* R^3*(epsilon[Dot]-epsilon[Matrix])/((epsilon[Dot]+2*epsilon [Matrix]));
ELECTROSTATIC DISPLACEMENT VECTOR FIELD IN POLAR AND CARTESIAN COORDINATES In this subsection, we determine expressions of electric displacement field inside and outside dielectric sphere in spherical coordinates as well as in cartesian rectangular coordinates. > D_Dot := - epsilon[Dot]*Gradient(V_Dot( r, theta)); D_Dot_r := DotProduct(V1, D_Dot); D_Dot_theta := DotProduct(V2, D_Dot); D_Dot_x := simplify(D_Dot_r*sin(theta) + D_Dot_theta* cos(theta)); D_Dot_x := subs([r=sqrt(x^2+z^2), cos(theta)= z/sqrt(x^2+z^2), sin(theta)=x/sqrt(x^2+z^2)],D_Dot_x); D_Dot_z := simplify(D_Dot_r*cos(theta) - D_Dot_theta*sin (theta)); D_Dot_z := subs([r=sqrt(x^2+z^2), cos(theta)=z/sqrt (x^2+z^2), sin(theta)=x/sqrt(x^2+z^2)],D_Dot_z); D_Matrix := - epsilon[Matrix]*Gradient(V_Matrix( r, theta)); D_Matrix_r := DotProduct(V1, D_Matrix); D_Matrix_theta := DotProduct(V2, D_Matrix); D_Matrix_x := simplify(D_Matrix_r*sin(theta) + D_Matrix_theta*cos(theta)); D_Matrix_x := subs([r=sqrt(x^2+ z^2), cos(theta)=z/sqrt(x^2+z^2), sin(theta)=x/sqrt(x^2+z^2) ],D_Matrix_x); D_Matrix_z := simplify(D_Matrix_r*cos(theta) D_Matrix_theta*sin(theta)); D_Matrix_z := subs([r=sqrt(x^2+ z^2), cos(theta)=z/sqrt(x^2+z^2), sin(theta)=x/sqrt(x^2+z^2) ],D_Matrix_z);
POLARIZATION VECTOR FIELD IN POLAR AND CARTESIAN COORDINATES In this subsection, we determine expressions of polarization field inside and outside dielectric sphere in spherical coordinates as well as in cartesian rectangular coordinates. > P_Dot := - (epsilon[Dot]-epsilon[0])*Gradient(V_Dot( r, theta)); P_Dot_r := DotProduct(V1, P_Dot); P_Dot_theta := DotProduct(V2, P_Dot); P_Dot_x := simplify(P_Dot_r*sin(theta) + P_Dot_theta*cos (theta)); P_Dot_x := subs([r=sqrt(x^2+z^2), cos(theta)=z/sqrt (x^2+z^2), sin(theta)=x/sqrt(x^2+z^2)],P_Dot_x); P_Dot_z := simplify(P_Dot_r*cos(theta) - P_Dot_theta*sin(theta)); P_Dot_z := subs([r=sqrt(x^2+z^2), cos(theta)=z/sqrt(x^2+z^2), sin(theta)=x/sqrt(x^2+z^2)],P_Dot_z); P_Matrix := - (epsilon [Matrix]-epsilon[0])*Gradient(V_Matrix( r, theta)); P_Matrix_r := DotProduct(V1, P_Matrix); P_Matrix_theta := DotProduct(V2, P_Matrix); P_Matrix_x := simplify(P_Matrix_r* sin(theta) + P_Matrix_theta*cos(theta)); P_Matrix_x := subs( [r=sqrt(x^2+z^2), cos(theta)=z/sqrt(x^2+z^2), sin(theta)= x/sqrt(x^2+z^2)],P_Matrix_x); P_Matrix_z := simplify(P_Matrix_r*cos(theta) P_Matrix_theta*sin(theta)); P_Matrix_z := subs([r=sqrt(x^2+ z^2), cos(theta)=z/sqrt(x^2+z^2), sin(theta)=x/sqrt(x^2+z^2) ],P_Matrix_z);
VOLUME AND SURFACE CHARGE In this subsection, we determine expressions of volume densities of fictive (virtual) electrostatic charge inside and outside the sphere as well as surface distribution of fictive charge at the sphere surface. > rho[Inside] := Divergence(P_Dot); rho[Outside] := simplify (Divergence(P_Matrix)); sigma[Surface] := simplify(DotProduct (V1, P_Dot));
PARTICULARIZATION TO A CASE STUDY: SILICON SPHERE IN VACUUM We particularize to the case of Silicon sphere with a radius R=10 cm in vacuum and submitted to a uniform electric field F=100000V/m. > epsilon[0] := evalf(Constant(permittivity_of_vacuum)); epsilon [Matrix] := evalf(Constant(permittivity_of_vacuum)); epsilon [Dot] := simplify(evalf(11.9*Constant(permittivity_of_vacuum))); R := 10*10^(-2); F_0 := 100000; V_Dot( r, theta) := V_Dot( r, theta); V_Matrix( r, theta) := V_Matrix( r, theta); E_Dot := E_Dot; E_Dot_r := E_Dot_r; E_Dot_theta := E_Dot_theta; E_Dot_x := E_Dot_x; E_Dot_z := E_Dot_z; E_Matrix := E_Matrix; E_Matrix_r := E_Matrix_r; E_Matrix_theta := E_Matrix_theta ; E_Matrix_x := E_Matrix_x; E_Matrix_z := E_Matrix_z; D_Dot := D_Dot; D_Dot_r := D_Dot_r; D_Dot_theta := D_Dot_theta; D_Dot_x := D_Dot_x; D_Dot_z := D_Dot_z; D_Matrix := D_Matrix; D_Matrix_r := D_Matrix_r; D_Matrix_theta := D_Matrix_theta; D_Matrix_x := D_Matrix_x; D_Matrix_z := D_Matrix_z; P_Dot := P_Dot; P_Dot_r := P_Dot_r; P_Dot_theta := P_Dot_theta; P_Dot_x := P_Dot_x; P_Dot_z := P_Dot_z; P_Matrix := P_Matrix; P_Matrix_r := P_Matrix_r; P_Matrix_theta := P_Matrix_theta; P_Matrix_x := P_Matrix_x; P_Matrix_z := P_Matrix_z; P_0 := P_0; rho[Inside] := rho[Inside]; rho[Outside] := rho[Outside]; sigma [Surface] := sigma[Surface]; sigma[0] := evalf(sqrt(P_Dot_r^2 + P_Dot_theta^2));
VISUALIZATION OF EQUIPOTENTIAL LINES The two following figures display respectively equipotentials in meridian plane (x0z) in spherical coordinates and in rectangular coordinates, red lines correspond to higher values of V and violet lines correspond to lower values of V. > V_Dot_Matrix_r_theta := piecewise( r^2 >= R^2, V_Matrix( r, theta), r^2 V_Dot_Matrix_x_z := subs([r = sqrt(x^2+z^2),cos(theta) = z/sqrt (x^2+z^2)], piecewise( r^2 >= R^2, V_Matrix( r, theta), r^2 f := subs([r=sqrt(x^2+z^2), cos(theta)=z/sqrt(x^2+z^2)],V_Dot( r, theta)): g := subs([r=sqrt(x^2+z^2), cos(theta)=z/sqrt(x^2+ z^2)],V_Matrix( r, theta)): V_Dot_Matrix := piecewise( (x^2+z^2) >= R^2, g, (x^2+z^2) implicitplot(x^2+z^2 = R^2, x = -R .. R, z = -R .. R, thickness = 1, color =blue): fig3 :=%: spacecurve([R*cos(theta), R*sin (theta), 0], theta = 0 .. 2*Pi, color = blue, thickness = 1): fig4 := %: display({fig1, fig4});
The following figure displays contourplot of scalar electrostatic potential, dark green lines correspond to higher values of V and pale green lines correspond to lower values of V. > display({fig2, fig3});
VISUALIZATION OF ELECTROSTATIC VECTOR FIELD LINES In the following figure, F fieldplot is added. > E_Dot_Matrix_x := Heaviside(-R^2+(x^2+z^2))*E_Matrix_x + Heaviside(R^2-(x^2+z^2))*E_Dot_x: E_Dot_Matrix_z := Heaviside(R^2+(x^2+z^2))*E_Matrix_z + Heaviside(R^2-(x^2+z^2))*E_Dot_z: fieldplot([E_Dot_Matrix_x, E_Dot_Matrix_z], x = -2*R .. 2*R, z = -2*R .. 2*R, grid=[8, 10],thickness=1,color=red,arrows=THICK, axes=BOXED,scaling=constrained): fig5 := %: display({fig2, fig3, fig5});
> implicitplot( [seq(sqrt(x^2+z^2)-(((1.2/0.01/K)^(-1))*(abs(x) /sqrt(x^2+z^2))^(-1))*piecewise(R^2-(x^2+z^2) >= 0, 1, 0),K=1. .12)], x = -2*R .. 2*R, z = -2*R .. 2*R, color = red, axes= boxed, scaling=constrained, linestyle=1, grid = [200, 200], font = [TIMES, BOLD, 16]): fig6 := %: implicitplot( [seq(sqrt(x^2+ z^2)/(F_0*(sqrt(x^2+z^2))^3+2*P_0)-(((4.223237078*K/1)^(-2))* (x/sqrt(x^2+z^2))^2)*piecewise(-R^2+(x^2+z^2) >= 0, 1, 0),K=1. .20)], x = -2*R .. 2*R, z = -2*R .. 2*R, color = red, axes= boxed, scaling=constrained, linestyle=1, grid = [200, 200], font = [TIMES, BOLD, 16]): fig7 := %: display({fig6, fig7});
> display({fig2, fig3, fig5, fig6, fig7});
The following figure presents electrostatic field lines as well as equipotential lines. > display({fig2, fig3, fig6, fig7});
VISUALIZATION OF ELECTROSTATIC DISPLACEMENT VECTOR FIELD LINES In the following figure, D fieldplot is added. > D_Dot_Matrix_x := Heaviside(-R^2+(x^2+z^2))*D_Matrix_x + Heaviside(R^2-(x^2+z^2))*D_Dot_x: D_Dot_Matrix_z := Heaviside(R^2+(x^2+z^2))*D_Matrix_z + Heaviside(R^2-(x^2+z^2))*D_Dot_z: fieldplot([D_Dot_Matrix_x, D_Dot_Matrix_z], x = -2*R .. 2*R, z = -2*R .. 2*R, grid=[10, 10],thickness=1,color=red,arrows=THICK, axes=BOXED,scaling=constrained): fig6 := %: display({fig2, fig3, fig6});
VISUALIZATION OF POLARIZATION LINES In the following figure, P fieldplot is added. > P_Dot_Matrix_x := Heaviside(-R^2+(x^2+z^2))*P_Matrix_x + Heaviside(R^2-(x^2+z^2))*P_Dot_x: P_Dot_Matrix_z := Heaviside(R^2+(x^2+z^2))*P_Matrix_z + Heaviside(R^2-(x^2+z^2))*P_Dot_z: fieldplot([P_Dot_Matrix_x, P_Dot_Matrix_z], x = -2*R .. 2*R, z = -2*R .. 2*R, grid=[10, 10],thickness=1,color=red,arrows=THICK, axes=BOXED,scaling=constrained): fig7 := %: display({fig2, fig3, fig7});
PLOT OF SURFACE ELECTROSTATIC CHARGE DISTRIBUTION The following figure presents surface charge distribution as function of azimuthal angle. > plot([sigma[Surface], 0], theta = 0..Pi, axes = boxed, color = red, font = [TIMES, BOLD, 16], labels = [theta, "Surface distribution"], labelfont = [16, 16], labeldirections = [horizontal, vertical] );
> Q[Surface] := int(int(sigma[Surface]*R*sin(theta)*R,theta = 0.. Pi),phi = 0 .. 2*Pi);
CONCLUSION In the present worksheet, we solve Poisson equation in the case of dielectric sphere surrounded by dielectric host matrix and submitted to uniform electrostatic field. We determine analytical expressions of scalar electrostatic potential, electrostatic field, electric displacement field and polarization field inside and outside the sphere. We use Maple graphical abilities to draw equipotential lines, electrostatic field lines, electrostatic fieldplot, electric displacement fieldplot and polarization fieldplot.
REFERENCES [1] Alexei V. Tikhonenko, Two-dimensional partial elliptic differential
equations in Maple, Maple Application Center, May 29, 2007. https://fr.maplesoft.com/applications/view.aspx?SID=4972 [2] Shun Lien Chuang, Physics of Optoelectronic Devices, John Wiley and Sons, New York, 1995. Legal Notice: © EL MAHDI ASSAID 2017. Maplesoft and Maple are trademarks of Waterloo Maple Inc. Neither Maplesoft nor the authors are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the authors for permission if you wish to use this application in for-profit activities.
