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Surface charge density of a dielectric particle near a plane
Journal o[Electrostatics, 20 (1987) 239-242
239
Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
Short Communication SURFACE CHARGE DENSITY OF A DIELECTRIC PARTICLE NEAR A PLANE
CAI XIAOPING
Department of Applied Chemistry, Shanghai Jiao Tong University, Shanghai (P.R. China) (ReceivedJune 5, 1987; accepted in revised form September 3, 1987)
The electrostatic problem of a charged dielectric particle near a conducting plane in the presence of a uniform field is encountered in a variety of engineering problems: microdischarge, electrostatic filters, and adhesion. The problem can be solved in a similar way as proposed previously if the charge distribution is known [ 1 ]. It is, however, very difficult to determine the charge distribution of the particle. In this paper, we calculate the charge density of a dielectric particle with a charge Q distributed over its surface only. Consider a dielectric particle of radius r with dielectric constant ~, which is a distance d away from an infinite conducting plane at zero potential, with the electric field far from the particle being normal to the plane and constant in magnitude, as indicated in Fig. 1. Let the charge density on the surface of the particle be a(t/). Let the conducting plane be located at z = 0, and the centre of the particle on the z-axis at z = d + r. The non-vanishing asymptotic electric field component is Ez = Eo for z > 0. The potential inside and outside the particle are denoted as Van and Vout, respectively. Both inside and outside the particle there are no free charges. Because of the axial symmetry, the problem can be written, in bispherical coordinates, as O(
sin.
3Vin~
tcosh G os,
O ['
sin.
OVin~
+ cosh
OYout~ 0___( sin~/ O~\cosh ~ - c o s ~/ O~ ]
0
0
]= OVout~
sin, t/
0. ]=0
(1) (2)
Yout Iz--*~ =Eoa sinh ~/(cosh C-cos t/)
(3)
Yout [~=0 =0
(4)
Vou~I¢=~o = V,n I~=~o 0Vout c OVin],
(5)
0~
a(17 ) a ¢=eo-- -~-Ie=eo-- eo cosh Co-COS ~/
0304-3886/87/$03.50
© 1987 Elsevier Science Publishers B.V.
(6)
240
dielectric particle
V
I Vout
d
grounded plane Fig. 1. Charged dielectric particle of radius r and dielectric constant e near a grounded plane with a normal electric field Eo far from the particle. where ~0 is the coordinate of the surface of the particle defined as ,o = l n [
(d2+2rd)~+d 1 ( d2 + 2rd ) ½
and
a= (d2 +2rd) ½ We take the solution to be of the form: Vin = (cosh ~ - cos r/) ½ ~ x. e -~"+½~ P~ (cos ~/)
(7)
n~0
Yout =
( c o s h ~ - c o s r / ) ½ ~ yn[e(n+½~¢-e -~"+½~1 Pn(cos r/) n~O
+Eoa sinh ~ / ( c o s h ~ - c o s r/)
(8)
and the surface charge density a(r/) to be of the form a(r/) = (cosh ~o-COS r/) 3/2 ~ z~P.(cos r/)
(9)
n ~--~0
where Pn (cos t/) is the Legendre polynomial of order n. T h e set of coefficients x., y . and zn are presented as vectors X, Y and Z for the convenience of writing. With the recurrence formula of Legendre polynomials
241
( 2 n + l ) c o s ~] Pn(cos t]) = ( n + l ) P . + l (cos ~7) +nPn_l(cos ~) and the boundary conditions at the surface of the particle, a system of linear algebraic equations is obtained as follows
AX+ BY=C
(10)
DX+ EY=FZ+G
(11)
where A = ( aij) , B = ( bij) , C= ( ci) , D = ( dij) , E = ( eij) , F = (fij) and G = (gi) are given as follows: aij= ( 0 e - (i+ ½)¢°
j----i otherwise
bij= {o(i+½)~°-e-(i+½)~°
j=i otherwise
Ci=
-8½ Eoa( i+ ½)e -(i+½~¢°
f --e½ie-(i-½)¢°
j=i-1
J e[ ½ie-(i- ~)¢o+ ½(i+ 1)e- (i+a/2)~o] dij -~ ~ e1(i+l~e-.+3/2)¢o 0 ~
j=i j=i+l otherwise
r - ½i[e(i-t)~°+e-(i-½)Co] | ½i[e (/- ~)~°+e-(i-~)¢o1
j=i-1
{ +½(i+l)[e(i+3/2){°+e-(i+3/2)~°] eij= I !(i+l)fe(i+3/2)~O+e-(i+3/e)~ol
j=i j=i+l otherwise
-ai/(2i-1)eo acosh ~o/eo oa(i+l)/(2i+3)Eo
j=i-1 j=i j=i+l otherwise
gi= -2½ Eoa[ ( i+ l )e-(i+3/2)¢O-ie-(i-½ )~o] The energy of the charge distribution is
U=fa(l?) Vout I~=~od S = 2 7 t a 2 Z T ( K + L Y ) = 2~a2Z T [ K + L ( E - D A - 1 B ) - 1( G - D A -
+L(E-DA-'B)
-1FZ]
1C)
(12)
242
2.0
1.5
d/r'=-1,Eo=2~: ~ d/r=1, EO--~ " ~ d/r=lO,Eo=T
o.5 ]_
0.0
0
_
90
,
f
=
120
180
Fig. 2. Relationship between the surface charge density a of the particle and the polar angle8 for
various valuesof d/r and Eo, ( ao = Q/4~r 2, ~ = ao/~o). where K = (ki) and L = (lij) are given by
ki = 8 ~Eoae-(i+½)~o
{ [e(i+t)~O_e-(i+½){o]/(i+½)
lij"~" 0
j=i otherwise
In equilibrium, the energy of the charge distribution must be minimum under the condition
Q= fa(rl)dS= 2~a2ZTM
(13)
w h e r e M = (mi)
mi =2½e-(i+t)¢°/ ( i + ½) With the Lagrangian multiplier method the surface charge density can be obtained. In order to get an approximate solution, some finite terms of z, are truncated in eqns. (12) and (13). In the case of truncating 20 terms of z, and taking e = 3 the surface charge density is shown in Fig. 2 for various values of d/r and Eo. References 1 Cai Xiaoping, The electrostatic problem of a dielectric sphere near a plane, J. Electrostatics, 19 (1987) 201-204.
239
Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
Short Communication SURFACE CHARGE DENSITY OF A DIELECTRIC PARTICLE NEAR A PLANE
CAI XIAOPING
Department of Applied Chemistry, Shanghai Jiao Tong University, Shanghai (P.R. China) (ReceivedJune 5, 1987; accepted in revised form September 3, 1987)
The electrostatic problem of a charged dielectric particle near a conducting plane in the presence of a uniform field is encountered in a variety of engineering problems: microdischarge, electrostatic filters, and adhesion. The problem can be solved in a similar way as proposed previously if the charge distribution is known [ 1 ]. It is, however, very difficult to determine the charge distribution of the particle. In this paper, we calculate the charge density of a dielectric particle with a charge Q distributed over its surface only. Consider a dielectric particle of radius r with dielectric constant ~, which is a distance d away from an infinite conducting plane at zero potential, with the electric field far from the particle being normal to the plane and constant in magnitude, as indicated in Fig. 1. Let the charge density on the surface of the particle be a(t/). Let the conducting plane be located at z = 0, and the centre of the particle on the z-axis at z = d + r. The non-vanishing asymptotic electric field component is Ez = Eo for z > 0. The potential inside and outside the particle are denoted as Van and Vout, respectively. Both inside and outside the particle there are no free charges. Because of the axial symmetry, the problem can be written, in bispherical coordinates, as O(
sin.
3Vin~
tcosh G os,
O ['
sin.
OVin~
+ cosh
OYout~ 0___( sin~/ O~\cosh ~ - c o s ~/ O~ ]
0
0
]= OVout~
sin, t/
0. ]=0
(1) (2)
Yout Iz--*~ =Eoa sinh ~/(cosh C-cos t/)
(3)
Yout [~=0 =0
(4)
Vou~I¢=~o = V,n I~=~o 0Vout c OVin],
(5)
0~
a(17 ) a ¢=eo-- -~-Ie=eo-- eo cosh Co-COS ~/
0304-3886/87/$03.50
© 1987 Elsevier Science Publishers B.V.
(6)
240
dielectric particle
V
I Vout
d
grounded plane Fig. 1. Charged dielectric particle of radius r and dielectric constant e near a grounded plane with a normal electric field Eo far from the particle. where ~0 is the coordinate of the surface of the particle defined as ,o = l n [
(d2+2rd)~+d 1 ( d2 + 2rd ) ½
and
a= (d2 +2rd) ½ We take the solution to be of the form: Vin = (cosh ~ - cos r/) ½ ~ x. e -~"+½~ P~ (cos ~/)
(7)
n~0
Yout =
( c o s h ~ - c o s r / ) ½ ~ yn[e(n+½~¢-e -~"+½~1 Pn(cos r/) n~O
+Eoa sinh ~ / ( c o s h ~ - c o s r/)
(8)
and the surface charge density a(r/) to be of the form a(r/) = (cosh ~o-COS r/) 3/2 ~ z~P.(cos r/)
(9)
n ~--~0
where Pn (cos t/) is the Legendre polynomial of order n. T h e set of coefficients x., y . and zn are presented as vectors X, Y and Z for the convenience of writing. With the recurrence formula of Legendre polynomials
241
( 2 n + l ) c o s ~] Pn(cos t]) = ( n + l ) P . + l (cos ~7) +nPn_l(cos ~) and the boundary conditions at the surface of the particle, a system of linear algebraic equations is obtained as follows
AX+ BY=C
(10)
DX+ EY=FZ+G
(11)
where A = ( aij) , B = ( bij) , C= ( ci) , D = ( dij) , E = ( eij) , F = (fij) and G = (gi) are given as follows: aij= ( 0 e - (i+ ½)¢°
j----i otherwise
bij= {o(i+½)~°-e-(i+½)~°
j=i otherwise
Ci=
-8½ Eoa( i+ ½)e -(i+½~¢°
f --e½ie-(i-½)¢°
j=i-1
J e[ ½ie-(i- ~)¢o+ ½(i+ 1)e- (i+a/2)~o] dij -~ ~ e1(i+l~e-.+3/2)¢o 0 ~
j=i j=i+l otherwise
r - ½i[e(i-t)~°+e-(i-½)Co] | ½i[e (/- ~)~°+e-(i-~)¢o1
j=i-1
{ +½(i+l)[e(i+3/2){°+e-(i+3/2)~°] eij= I !(i+l)fe(i+3/2)~O+e-(i+3/e)~ol
j=i j=i+l otherwise
-ai/(2i-1)eo acosh ~o/eo oa(i+l)/(2i+3)Eo
j=i-1 j=i j=i+l otherwise
gi= -2½ Eoa[ ( i+ l )e-(i+3/2)¢O-ie-(i-½ )~o] The energy of the charge distribution is
U=fa(l?) Vout I~=~od S = 2 7 t a 2 Z T ( K + L Y ) = 2~a2Z T [ K + L ( E - D A - 1 B ) - 1( G - D A -
+L(E-DA-'B)
-1FZ]
1C)
(12)
242
2.0
1.5
d/r'=-1,Eo=2~: ~ d/r=1, EO--~ " ~ d/r=lO,Eo=T
o.5 ]_
0.0
0
_
90
,
f
=
120
180
Fig. 2. Relationship between the surface charge density a of the particle and the polar angle8 for
various valuesof d/r and Eo, ( ao = Q/4~r 2, ~ = ao/~o). where K = (ki) and L = (lij) are given by
ki = 8 ~Eoae-(i+½)~o
{ [e(i+t)~O_e-(i+½){o]/(i+½)
lij"~" 0
j=i otherwise
In equilibrium, the energy of the charge distribution must be minimum under the condition
Q= fa(rl)dS= 2~a2ZTM
(13)
w h e r e M = (mi)
mi =2½e-(i+t)¢°/ ( i + ½) With the Lagrangian multiplier method the surface charge density can be obtained. In order to get an approximate solution, some finite terms of z, are truncated in eqns. (12) and (13). In the case of truncating 20 terms of z, and taking e = 3 the surface charge density is shown in Fig. 2 for various values of d/r and Eo. References 1 Cai Xiaoping, The electrostatic problem of a dielectric sphere near a plane, J. Electrostatics, 19 (1987) 201-204.