New solutions for charge distribution on conductor surface

Journal of Electrostatics 77 (2015) 147e152

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New solutions for charge distribution on conductor surface Peter A. Polyakov a, b, Natalia E. Rusakova a, *, Yulia V. Samukhina a a b

Department of Physics, M.V. Lomonosov Moscow State University, Leninskie Gory, Moscow, 119991, Russia Plekhanov Russian University for Economics, Stremyanny per. 36, Moscow, 117997, Russia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 October 2014 Received in revised form 13 July 2015 Accepted 10 August 2015 Available online 28 August 2015

In the paper, a problem of electrostatics for charge distribution on a conductor surface is analytically solved for three new particular cases of conducting surfaces with complicated shape and specified value of electrostatic potential. The exact analytical expressions for surface charge density for the bodies are obtained. All the solutions are represented in a clear view of 3D graphs. It is shown that the proposed method of electrostatic problem for conductors allows to obtain infinitely many numerical solutions for the problem but only several special cases can be solved analytically. © 2015 Elsevier B.V. All rights reserved.

Keywords: Charge distribution Analytical solution Problem of electrostatics Conductor

1. Introduction Let us consider a charged conducting body with an arbitrary shape in vacuum. One can obtain a uniquely determined expression for electrostatic potential f governed with the body charge in a space point by solving Dirichlet problem for Laplace equation [1]:

D4 ¼ 0;

(1)

4jS ¼ 41 ;

(2)

4j∞ ¼ 0

(3)

where 41 is a certain constant. According to Equation (3), the electric potential at infinity is chosen to be zero. The problem has a set of analytical solutions and most of them are given in classical electrodynamics textbooks [2e5], e.g. a charged conducting ellipsoid and its particular cases. There are analytical solutions that represent a surface of two intersecting spheres [6]. A class of solutions derived by means of electrostatic image method and in terms of complex potential [7,8] exists. There are some elegant solutions for a uniformly charged elliptic ring [9], two conducting spheres [10], a uniformly charged square [11] and for a uniformly charged rectangular shape [12]. The mentioned

* Corresponding author. E-mail address: [email protected] (N.E. Rusakova). http://dx.doi.org/10.1016/j.elstat.2015.08.003 0304-3886/© 2015 Elsevier B.V. All rights reserved.

analytical solutions are of great concern for electrostatics problem since they enable one to analyze efficiency of its different numerical solutions and serve as a beacon for qualitative understanding of electrostatic charge distribution on a surface of different conducting shapes. A new class of non-trivial solutions for the electrostatics of conductors is investigated in the paper. It is derived from a wellknown presentation of a solution for the Laplace equation as an expansion in terms of spherical harmonics [13]:

4ðr; q; fÞ ¼

þ∞ X n X ank k $Y ðq; fÞ; nþ1 n r n¼0 k¼n

(4)

where Ynk ðq; fÞ are spherical harmonics. In particular, Lord Rayleigh used such expansion of a potential in series of spherical harmonics to investigate an instability of a charged drop with an arbitrarily deformed surface [14]. Every term of the infinite series

n X ank k $Y ðq; fÞ nþ1 n r k¼n

is a particular solution of the Laplace equation [15]. And any finite sum of the series (5) is also a solution for the Laplace equation

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4ðr; q; fÞ ¼

N X n X ank k $Y ðq; fÞ nþ1 n r n¼0 k¼n

(5)

One can use the fact in order to obtain new analytical solutions for the electrostatics problems of conductors. 2. The problem-solving method In order to derive an analytical solution, let us choose such special shape of a charged conducting surface S that coincides with one of the equipotential surfaces governed with Equation (5) at a constant potential value 4S. A solution for the outer boundary Dirichlet problem (1)e(3) will be obtained when one fits coefficients ank such as the potential (5) is equal to the specified value 4S in any point of the closed conducting surface S i.e.

4S ¼

N X n X ank k $Y ðq; fÞ nþ1 n r n¼0 k¼n

(6)

Expression (6) is the (N þ 1)th degree polynomial with respect to reciprocal radius 1/r. Thus, it is necessary to solve a polynomial equation with the coefficients as a linear combination of spherical functions Ynk ðq; fÞ to obtain the feasible shape of the closed equipotential surface. Any solution is a function of spherical coordinates q, f and governs an equipotential surface

r ¼ rðq; fÞ:

(7)

There are N þ 1 different solutions in a general way. But the real solution governing a closed surface for specified coefficients ank is single in accordance with a uniqueness of the solution for the electrostatics problem. Mathematical demonstration of the fact is a separate issue for the polynomials with the specified coefficients. It should be pointed out that an analytical solution for the Equation (6) exists for polynomials solvable by quadratures i.e. up to the 4th degree only when N  3. Otherwise the roots are transcendental and a numerical solution is available only. Thus there are four sets of possible analytical solutions for Dirichlet problem (1)e(3) for closed shapes derived with Equation (6). A surface of the shapes (7) is specified with analytical expressions for solutions of respective polynomial equations. When the potential depends on polar angle q only and does not on azimuthal angle f (in axially symmetric case) the derived surfaces are surfaces of revolution with respect to Z-axis. They are determined with the following equation [15].

4S ðr; qÞ ¼

N X n¼0

an

Pn ðcos qÞ r nþ1

(8)

40 ¼

3. The surface equation and the charge distribution in case with N ¼ 1

Jx2  xHk1 P1 ðcosqÞ ¼ 0;

(9)

where ε0 is the dielectric constant, a1 and a0 are arbitrary constants,

(10)

(11)

where

k1 ¼ a1 =ða0 r0 Þ

(12)

is a dimensionless coefficient. The surface shapes obtained from Equation (11) when one choose signs «þ» and «» are congruous by reflection. Let us solve Equation (11) with sign «». Since x(q) is an absolute value of a dimensionless radius-vector, only its nonnegative real values are physically meaningful. The Equation (11) has two roots but only one is physically correct. Thus, we obtain the following equation for the surface in terms of dimensionless spherical coordinate x (10) derived above:

xðqÞ ¼

1þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4Jk1 cos q ; f2½0; 2p; 2J

(13)

where Jk1 is to satisfy the following relation

Jk1  1=4:

(14)

In a case when a value of Jk1 does not satisfy the relation (14), x(q) is a complex value for some q and the resulting surface has discontinuities. Let us investigate the charge distribution on the surface (13). It is well known that the electric field from a conductor close to its surface equals

!  s ¼  V 4; ε0

(15)

where s is the surface charge density. Let us use the following notation for the surface charge density on a spherical conductor with radius r0 and total charge q ¼ a0 as in Equation (11)

s0 ¼

Let us consider the case when N ¼ 1 in the series (8). Then there are only two terms in the expression for the electrostatic potential

a0 4 4S ; x ¼ r=r0 ; J ¼ S ¼ 4pε0 r0 40 a0 =4pε0 r0

for the electrostatic potential of a conducting sphere with radius r0 and total charge q ¼ a0, the radius-vector normalized with the sphere radius and the dimensionless potential respectively. Let us divide the two sides of Equation (9) by potential 40. Then surface Equation (9) takes the following dimensionless form:

E¼

Where N  3, Pn(cosq) are the nth degree Legendre polynomials, an ¼ an0 in series (5). In particular, it is a uniformly charged sphere when N ¼ 0. Let us thoroughly examine some particular cases with N ¼ 1 and N ¼ 2.

  1 a0 a1 ± 2 P1 ðcos qÞ ¼ 4S ; f2½0; 2p; 4pε0 r r

4S is a constant potential on the shape surface. Parameter a0 has the meaning of a total electric charge q distributed on the surface specified with Equation (9). In this case, the conductor surface and the surface charge distribution are represented as surfaces of revolution about the Z-axis though non-symmetrical with respect to the X and Y axes as it is shown below. In order to rewrite Equation (9) in a dimensionless form let us use the following notations

a0 4pr02

(16)

Then the surface charge density derived from Equation (15) can be rewritten in a dimensionless form

s ~¼ s ¼ s0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2   dJ 1 dJ 2 þ 2$ dx dq x

(17)

P.A. Polyakov et al. / Journal of Electrostatics 77 (2015) 147e152

σ sinθ

Then the following expression for the dimensionless surface charge density distribution is obtained in the above terms.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     1 2k1 cosq 2 k1 sinq 2 ~¼ s þ þ 2 3 3 x x x

149

1.0

(18)

0.5

Let us use the following coordinate system to plot the surface shape and the charge distribution:

8 < X ¼ x sinq sin4 Y ¼ x sinq cos4 : Z ¼ x cosq

(19)

All 3D plots below are represented in Cartesian coordinate system in Equation (19). The cross-sections with the plane passing through z axis perpendicular to Oxy coordinate plane for the surface shape and the surface charge distribution for the shape with values J¼1 and k1 ¼ 0.25 close to critical values determined by Equation (14) are plotted in Fig. 1 and Fig. 2. The 3D plots for the surface shape and the surface charge distribution with J ¼ 1 and k1 ¼ 0.25 are shown in Fig. 3 and Fig. 4.

0.5

0.5

1.0

σ cosθ

0.5

1.0 Fig. 2. The charge distribution (18) on the surface (13).

4. Equations for the surface and the surface charge distribution in case with N ¼ 2 Let us consider the case when N ¼ 2 and a1 ¼ 0 in the series (8). Then the surface equation is

  1 a0 a2 ± 3 P2 ðcos qÞ ¼ 4: 4pε0 r r

(20)

Let us denote

. k2 ¼ a2 a0 r02

(21)

Then the surface equation takes the following dimensionless form in the above terms (10)

Jx3  x2 Hk2 P2 ðcos qÞ ¼ 0

(22)

а) Let us consider the case with sign “þ”:

Fig. 3. The cut view of the surface (13).

Jx3  x2 þ k2 P2 ðcos qÞ ¼ 0

(23)

The equation has three roots but two of them are imaginary and have no physical significance similar to Section 3. The real root is determined by

ξsinθ 1.0

0.5

0.5

0.5

1.0

ξcosθ

0.5

1.0 Fig. 1. The cross-section for the surface (13).

Fig. 4. The cut view for the surface charge distribution (18) for the shape (13).

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P.A. Polyakov et al. / Journal of Electrostatics 77 (2015) 147e152

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 1 1 $ 1 þ 1=3 x1 ðqÞ ¼ LðqÞ  L2 ðqÞ  4 3J 2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 !! þ LðqÞ þ L2 ðqÞ  4

ξ1 sinθ (24)

1.0

0.5

where

LðqÞ ¼ 2 

  27 2 J k2 $ 3 cos2 q  1 2

(25)

Though L (q)  4  0 for some values of polar angle q it is easy to prove that expression (24) is real. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi If L2 ðqÞ  4 is a real number then x1(q) is real too. In this case x1(q) is also nonnegative since solving the following inequality 2

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 1 1 $ 1 þ 1=3 LðqÞ  L2 ðqÞ  4 3J 2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 !! þ LðqÞ þ L2 ðqÞ  4 0

(27)

(28)

1  4 $ 1 þ 2 cos 3J 3

0.5

The surface shape and the surface charge density distribution with critical values k ¼ 0.25 and J ¼ 0.76 are plotted in crosssection perpendicular to Oxy coordinate plane in Fig. 5 and Fig. 6. The surface is a surface of revolution about Z-axis and is symmetrical with respect to all three coordinate planes Oxy, Oxz, Oyz as opposed to the case with N ¼ 1. The 3D Plots for the same functions are shown in Fig. 7 and Fig. 8.

(29) Jx3  x2  k2 P2 ðcos qÞ ¼ 0 (30)

So x1(q) is real for these values of L(q) also. The nonnegativeness condition is

4 1  3 2

(31)

2p  f  2p

(32)

cos

J k2  4=27

σ sinθ 0.6 0.4

where x1(q) is determined by Equation (24).

0.2

(33)

Thus, x1(q) is real and nonnegative for any polar angle q if the restriction (33) holds. Similar to Section 2 one can obtain the following expression for the dimensionless charge density on polar angle q, parameter k2 and the dimensionless potential J

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u 3k3 cos2 q  1 1 9k2 cos2 q sin2 q t ~¼ s þ  ; 2x41 ðqÞ x21 ðqÞ x81 ðqÞ

(35)

The equation has one real root and two imaginary ones. Thus we derive from Equation (35) the following equation for the surface in dimensionless spherical coordinates

so any value f from the possible range is satisfactory therefore the value L(q) is restricted with the relation (27) only. The following restriction can be obtained from the Equation (27) 2

ξ1 cosθ

b) Let us consider the case when there is sign “d“ before parameter k2 in Equation (11):

And this can be feasibly reduced to

x1 ðqÞ ¼

1.0

Fig. 5. The cross-section for the surface (24).

Then x1(q) can be rewritten as the following

  1=3  1=3  1 1 $ 1 þ 1=3 rei4 x1 ðqÞ ¼ þ rei4 3J 2

0.5

(26)

Let us use the following notation

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  4  L2 ðqÞ 2 2 : r ¼ L ðqÞ þ 4  L ðqÞ ¼ 2; 4 ¼ arctg LðqÞ

0.5

1.0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gives us L(q)  2 that is the condition for L2 ðqÞ  4 to be real. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Let us consider the case when L2 ðqÞ  4 is imaginary, i.e.

2 < LðqÞ < 2:

1.0

0.4

0.2

0.2

0.4

σ cosθ

0.2 0.4

(34)

0.6 Fig. 6. The cross-section for the charge distribution (34) on surface (24).

P.A. Polyakov et al. / Journal of Electrostatics 77 (2015) 147e152

151

ξ2 sinθ 1.0 0.5 1.5

1.0

0.5

0.5

1.0

1.5

ξ2 cosθ

0.5 1.0 Fig. 9. The cross-section for the surface (36).

σ sinθ 0.4 0.2 Fig. 7. The cut view of the surface (24).

0.6 x2 ðqÞ ¼

21=3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 3J MðqÞ þ M 2 ðqÞ  4  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 MðqÞ þ M 2 ðqÞ  4

1 þ 3J

þ

27 2 J k2 2

(37)

(38)

If the restriction (38) is not fulfilled the resulting surface has discontinuities.

Fig. 8. The cut view for the charge distribution (34) on the surface (24).

0.6

σ cosθ

Fig. 10. The cross-section for the charge distribution (39) on surface (36).

Similar to Section 2 one can feasibly prove that x2(q) is real and nonnegative for any polar angle q if the following restriction is satisfied

J2 k2  8=27

0.4

0.4

where

MðqÞ ¼ 2 þ

0.2

(36)

3,21=3 J

 3 cos2 q  1

0.2 0.2





0.4

Like the procedure mentioned above, we obtain the expression for dimensionless density of charge distribution on the surface of the rotational shape

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u 3k3 cos2 q  1 1 9k2 cos2 q sin2 q t ~¼ s þ þ ; 2x42 ðqÞ x22 ðqÞ x82 ðqÞ

(39)

where x2(q) is obtained from Equation (36). There are graphs for the dependencies of the dimensionless x2(q) (36) and surface charge density (39) with critical values of parameters k2 ¼ 0.5 and J ¼ 0.768 in Fig. 9 and Fig. 10. For all graphs shown below the J and k2 are also equal to the critical values. The 3D plots for the same functions are shown in Fig. 11 and Fig. 12.

Fig. 11. The cut view for the surface (36).

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revolution. For these surfaces exact analytical solutions can be obtained but in the paper the new analytical formulas were obtained for three particular cases only. Besides their theoretical significance the analytical expressions for charge distribution on these surfaces also have applied meaning, e.g. to analyze efficiency of different numerical solutions for the electrostatic problems.

References

Fig. 12. The cut view for the charge distribution (39) on surface (36).

5. Conclusion Thus, it is shown in the paper that some new solutions for electrostatic problem exist that are analytical solutions for the (N þ 1)th degree polynomial Equation (8). In general, an analytical solution is possible only for polynomials up to the 4th degree. It is also shown that it is possible to solve the electrostatic problem analytically for some new 3D conducting shapes. The detailed solutions are given for the three particular cases when the conducting shapes are the surfaces of revolution. The analytical formulas of the charge distribution on the surfaces are obtained. The features of the charge distribution are investigated and the appropriate graphs are plotted. In general, the method derives a set of complicated surfaces of

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