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Revista Facultad de Ingeniería ISSN: 0717-1072 [email protected] Universidad de Tarapacá Chile
Hernandes, J. A.; Oliveira Capelas de, E.; Assis, A. K. T. Potential, electric field and surface charges close to the battery for a resistive cylindrical shell carrying a steady longitudinal current Revista Facultad de Ingeniería, vol. 12, núm. 2, 2004, pp. 13-20 Universidad de Tarapacá Arica, Chile
Available in: http://www.redalyc.org/articulo.oa?id=11412202
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REVISTA FACULTAD DE INGENIERÍA, U.T.A. (CHILE), VOL. 12 Nº2, 2004, pp. 13-20
POTENTIAL, ELECTRIC FIELD AND SURFACE CHARGES CLOSE TO THE BATTERY FOR A RESISTIVE CYLINDRICAL SHELL CARRYING A STEADY LONGITUDINAL CURRENT J. A. Hernandes1
E. Capelas de Oliveira2
A. K. T. Assis3
Recibido el 12 de mayo de 2004, aceptado el 30 de agosto de 2004 RESUMEN En este trabajo consideramos una capa resistiva cilíndrica que transporta una corriente constante. Una batería genera la corriente en el centro del conductor. Estudiamos el comportamiento del potencial, campo eléctrico y cargas superficiales cerca de la batería. Palabras clave: Potencial eléctrico, potencial cerca de la batería, cargas superficiales. ABSTRACT In this work we consider a long, resistive cylindrical shell carrying a steady current. A battery in the middle of the wire generates the current. We study the behavior of the potential, electric field and surface charges close to the battery. Keywords: Electric potential, potential close to the battery, surface charges. INTRODUCTION There has been recently a great interest in the electric field outside resistive conductors carrying steady currents and a number of problems have been published in the literature: coaxial cables, [14], [8], [13], [1] and [2]; cylindrical shell with azimuthal current, [12], [10] and [16]; planes, [13]; twin leads, [15] and [4]; a long strip, [9]; and a long straight cylindrical conductor with longitudinal current, [5]. In this last case it has only been considered the region far from the battery. It was then found that the potential and surface charges vary linearly with the longitudinal component. Here we analyse the situation close to the battery in order to understand the behaviour of surface charges at a discontinuity in the potential. We consider a hollow cylindrical shell of radius a and >> a . The shell has an uniform length with resistivity and carries a steady current I along the positive z direction which coincides with its axis. We suppose an idealized linear battery at (ρ ,ϕ , z ) = (a,ϕ ,0 ) , see Fig. 1, in analogy with Heald's treatment which considered a “line” battery at (ρ ,ϕ , z ) = (a,π , z ) driving current azimuthally in a uniform cylindrical resistive 1
sheet, [10]. We suppose a vacuum inside and outside the shell.
Fig. 1 A resistive cylindrical shell of radius a and length carrying a steady current in the z direction. There is a linear battery in z = 0 generating a difference of potential of 2φ 0 . Ohm's law can be
written as J = gE , where
J = (I / 2πat )zˆ is the volume current density ( t 0) = (φ R − φ L ) + R + φ0 2
(1) (2) Fig. 3
The potential of Fig. 2 can be decomposed in two parts: the shell held at constant but discontinuous potentials (dashed lines), and the shell with a continuous potential varying linearly with the longitudinal coordinate (continuous line).
ELECTROSTATIC SOLUTION OF THE CYLINDRICAL SHELL WITH IS TWO HALVES HELD AT CONSTANT AND OPPOSITE POTENTIALS
Fig. 2 Potential along the shell. Our goal is to find the potential and electric field at ρ < a and at ρ > a given the boundary conditions above. Then we can obtain the surface charge densities at the inner and outer surfaces of the shell. This problem can be separated in two parts: a) the electrostatic situation of a cylindrical shell separated at z = 0 by a thin insulating barrier held at −φ 0 for z < 0 and at φ 0 for z > 0 ; and b) a continuous linear potential
Suppose two semi-infinite cylindrical shells, located at z < 0 and z > 0 , see Fig. 1. The shell at z < 0 is held at the constant potential −φ 0 , while the shell at z > 0 is held at φ 0 , [17]. We can solve Laplace's equation,
∇ 2φ = 0 , in cylindrical coordinates using separation of variables in the form φ jk (ρ ,ϕ , z ) = R jk (ρ )Φ j (ϕ )Z k (z ) , where the functions R jk , Φ j and Z k obey the equations: d 2 R jk dρ
2
+
1 dR jk 2 j 2 − k + 2 R jk = 0 , ρ dρ ρ d 2Φ j
φ (a,ϕ , z ) = (φ R − φ L ) z + (φ R + φ L ) 2 for − 2 ≤ z ≤ 2 , see Fig. 3. The first part has been partially solved in [7] and the second one in [5]. Our intention is to deepen these studies in order to understand the behaviour close to the battery.
dϕ 2 d 2Zk dz 2
− j 2Φ j = 0 ,
(3)
+ k 2Zk = 0
The final solution φ (ρ ,ϕ , z ) is a linear combination of all possible solutions φ jk (ρ ,ϕ , z ) .
Due to the rotational symmetry of the system, the solution will not depend on ϕ . This means j = 0 and
Φ j = constant . 14
Revista Facultad de Ingeniería, Chile, Vol. 12 Nº2, 2004
Potential, electric field and surface charges close to the battery for a resistive cylindrical shell carrying a steady longitudinal current
In this situation, the boundary conditions are antisymmetric about the coordinate, z φ (a,ϕ ,− z ) = −φ (a,ϕ , z ) , so the solution must also be anti-symmetric at all points: φ (ρ ,ϕ ,− z ) = −φ (ρ ,ϕ , z ) and φ (ρ ,ϕ ,0 ) = 0 . Additionally, we must have a limited solution in both and coordinates, z ρ
φ (ρ ,ϕ , z → ∞ ) ≤ φ 0 < ∞
and
This leads to a solution for
φ (ρ → ∞,ϕ , z ) → 0 . Zk
of the form:
Z k (z ) = sin (kz ) . The equation for R jk is the modified
Bessel equation, whose solutions are I j (kρ ) (regular
In Eq. (9), xn are the zeroes of the Bessel function of order zero, J 0 (x n ) = 0 , with n = 1,2,
SOLUTION FOR THE CYLINDRICAL SHELL WITH STEADY CURRENT AND BATTERY Eqs. (13) and (14) of [5] give the solution of a
conducting wire with radius a , length >> a and carrying a steady current as: z φ + φL φ (ρ ≤ a,ϕ , z ) = (φ L − φ R ) + R 2
for ρ → 0 and irregular for ρ → ∞ ) and K j (kρ ) (regular for ρ → ∞ and irregular for ρ → 0 ), [6].
( ρ) z φ (ρ ≥ a, ϕ , z ) = (φ L − φ R ) + ( a) φR + φL ( ρ ) + ( a) 2
The general solution has then the form: ∞
φ (ρ ≤ a,ϕ , z ) = ∫ Ak I 0 (kρ )sin (kz )dk
(4)
0
∞
φ (ρ ≥ a,ϕ , z ) = ∫ Bk K 0 (kρ )sin (kz )dk
(5)
0
where the coefficients Ak and Bk must be determined from the boundary conditions. Eqs. (4) and (5) can be seen as a sine Fourier transform of a function Ψ : ∞
2 Ψ (ρ , ϕ , k ) = φ (ρ , ϕ , z ) (kz )dz = π 0
∫
π 2 Ak I 0 (kρ ), ρ ≤ a = π 2 Bk K 0 (kρ ) ρ ≥ a
(12)
The solution of the problem with the battery is the sum of Eqs. (8) and (11) for the region inside the shell, and the sum of Eqs. (10) and (12) for the region outside the shell, namely:
φ (ρ ≤ a, ϕ , z ) =
2φ 0 π
∞
∫ 0
I 0 (kρ ) (kz ) dk + I 0 (ka ) k
(13)
φ + φL + (φ L − φ R ) + R 2 z
(6)
∞ 2φ 0 K 0 (kρ ) φ (ρ ≥ a, ϕ , z ) = π 0 K 0 (ka ) ( ρ) z φR + φL + (φ L − φ R ) + ( a) 2
∫
Calculating Eq. (6) in ρ = a and applying the boundary conditions φ (a,ϕ , z < 0 ) = −φ 0 and φ (a,ϕ , z > 0 ) = φ 0 yields the coefficients Ak and Bk : 2φ 0 2φ 0 Ak = , Bk = πkI 0 (ka ) πkK 0 (ka )
(11)
(kz ) dk +
k ( ρ) ( a)
(14)
Fig. 4 shows the equipotentials with φ L = φ 0 , φ R = −φ 0
(7 )
and
a = 10 .
The final solution can be written as (see Appendix):
φ (ρ ≤ a,ϕ , z ) =
2φ 0 π
∞
I 0 (kρ ) sin (kz ) dk k
∫ 0 I 0 (ka )
−x z a z 1 ∞ J 0 (xn ρ a ) e n −∑ z 2 n=1 J1 (xn ) xn 2φ 0 ∞ K 0 (kρ ) sin (kz ) dk φ (ρ ≥ a,ϕ , z ) = ∫ π 0 K 0 (ka ) k
= 2φ 0
(8) (9)
(10)
Revista Facultad de Ingeniería, Chile, Vol. 12 Nº2, 2004
15
J.A. Hernandes, E.Capelas de Oliveira, A.K.T. Assis
internal and external densities of surface charge as given by, respectively (where ε 0 = 8.85 × 10 −12 C 2 N −1m −2 ):
(
)
σ 2 32 3 a − , ϕ , z = − lim ε 0 E ρ (ρ < a, ϕ , z ) = = =
2φ 0 ε 0 π
∞
ρ →a
I (ka )
∫ I 10 (ka) sin(kz )dk = 0
2φ 0 ε 0 z a z
(
∞
∑ e −x
n
(17 )
z a
n =1
)
σ 5786 a + , ϕ , z = lim ε 0 E ρ (ρ > a, ϕ , z ) = = Fig. 4
Equipotentials of conducting wire close to z = 0 . This is a plot of Eqs. (13) and (14) with φ L = −φ R = φ 0 and a = 10 . There is a steady current flowing along the positive z direction.
The electric field E can be obtained from the potential utilizing the relation E = −∇φ . This yields:
2φ E (ρ < a, ϕ , z ) = − 0 π ∞ ∞ ( ) I (kρ ) " # ρˆ 1 (kz )dk + zˆ I 0 kρ ! (kz )dk + I (ka ) 0 I 0 (ka ) 0 0 φ −φ (15) + zˆ L R
∫
2φ 0 ε 0 π
∞
ρ →a
K1 (ka )
∫ K 0 (ka ) sin(kz )dk +
(18)
0
ε (φ − φ ) ε (φ + φ ) − 04 L 4 R z + 0 R 4 L a ln ( a ) 2a ln ( a ) Fig. 5 shows these normalized densities of surface charge as a function of z a . Both of them diverge to infinity when z a > 1 the internal density of surface charge goes to zero faster than the external one.
∫
.
E (ρ > a, ϕ , z ) =
2φ 0 ⋅ π
∞ ∞ K (kρ ) )+*-, (kz )dk − zˆ K 0 (kρ ) $&%( ' (kz )dk + ⋅ ρˆ 1 K 0 (ka ) 0 K 0 (ka ) 0
∫
∫
− 1 ρˆ (φ L − φ R ) /z − φ R + φ L + 0 1 / 2 ( a) ρ + 0 1 / − φ φ ( ) + zˆ L / R 0 1 / ρ ( a)
Fig. 5
(16)
The surface charge distribution σ (a,ϕ , z ) can be found by applying Gauss' law and choosing a gaussian surface surrounding a small piece of the conductor surface. In the limit of an infinitesimal surface this yields the 16
Densities of surface charge as a function of the z (longitudinal) coordinate for a resistive hollow cylindrical shell carrying a steady current. The dashed (continuous) lines is the normalized internal (external) density of surface charges. Both densities are normalized by the internal density ( ρ = a − ) at z = a
Revista Facultad de Ingeniería, Chile, Vol. 12 Nº2, 2004
Potential, electric field and surface charges close to the battery for a resistive cylindrical shell carrying a steady longitudinal current
DISCUSSION Jefimenko and Heald considered a similar geometry (resistive cylindrical shell of radius a coaxial with the z axis) but carrying a steady azimuthal current, [18], [10] and [16]. A line battery located at (ρ ,ϕ , z ) = (a,π , z ) had its terminals at potentials ±φ 0 . They found equal internal and external surface charge densities given by:
σ ; 0 z a ln ( a ) charge distribution is null, σ (a, z 0 ) = 0 . In the interval A 1 ≤ a ≤ 100 , equating Eq. (31) to zero yields
A
z 0 a = 0.0890 + 0.05068 a ,
A
for z 0 ≈ 0.507 a >> 1 . Although our solution is valid only for
A
or
z 0 . We choose a contour of the type shown in Fig. 6 for z > 0 . The contour integral can thus be divided in three parts: along the real k axis, along the path Cr and along the path C R . The integral along the path Cr is given by:
lim
r →0
∫
Cr
I 0 (kρ ) e ikz dk = I 0 (ka ) k
( ) ( )ire ϕ dϕ = −iπ = lim ∫ ( e ϕ a) re ϕ π I r 0
r →0
I 0 r e iϕ ρ exp ikre iϕ 0
i
Revista Facultad de Ingeniería, Chile, Vol. 12 Nº2, 2004
i
i
(34)
Potential, electric field and surface charges close to the battery for a resistive cylindrical shell carrying a steady longitudinal current
REFERENCES [1] A.K.T. Assis and J.I. Cisneros. The problem of surface charges and fields in coaxial cables and its importance for relativistic physics. Open Questions in Relativistic Physics, pp. 177–185. F. Selleri (editor), Apeiron, Montreal, pp.508-511, 1998.
Fig. 6
Contour to calculate the integral of Eq. (8), for z > 0 . For z < 0 we choose a symmetrical contour reflected at the horizontal (real k ) axis
The integral along the path C R is limited, and vanishes for R → ∞ : I (kρ ) e ikz
e ikz
(35)
0 dk ≤ lim ∫ dk = 0 ∫ R →∞ C I 0 (ka ) k R→∞ C k R R
lim
Using Cauchy's theorem, the integral I given by:
lim
∫
r →0 R →∞ C
I 0 (kρ ) e ikz dk = I 0 (ka ) k
∞
∫
−∞
I 0 (kρ ) e ikz dk − iπ = I 0 (ka ) k
I 0 (k n ρ ) e ikn z = 2iπ Res(k n ) = 2iπ I (k a ) ak n n =1 n =1 1 n ∞
∞
∑
∑
is
[2] A.K.T. Assis and J.I. Cisneros. Surface charges and fields in a resistive coaxial cable carrying a constant current. IEEE Transactions on Circuits and Systems I, 47:63–66, 2000. [3] A.K.T. Assis, J.A. Hernandes and J.E. Lamesa. Surface charges in conductor plates carrying constant currents. Foundations of Physics, 31:1501–1511, 2001. [4] A.K.T. Assis and A.J. Mania. Surface charges and electric field in a two-wire resistive transmission line. Revista Brasileira de Ensino de Física, 21:469–475, 1999. [5] A.K.T. Assis, W.A. Rodrigues Jr., and A.J. Mania. The electric field outside a stationary resistive wire carrying a constant current. Foundations of Physics, 29:729–753, 1999.
(36)
where Res(k n ) is the residue of the integrand in
[6] G.B. Arfken and H.J. Weber. Mathematical Methods for Physicists. Academic Press, San Diego, 4th edition, Chap. 11, 1995. [7] E. Butkov. Mathematical Physics. Wesley, Reading, pp. 403, 1968.
Addison-
k = k n = ixn a . In the equation above we utilized the (x ) = I1 (x ) involving modified Bessel relation I 0 '
[8] D.J. Griffiths. Introduction to Electrodynamics. Prentice Hall, New Jersey, 3rd edition, pp. 336337, 1999.
relation Jν (ix ) = i Iν (x ) ):
[9] J.A. Hernandes and A.K.T. Assis. The potential, electric field and surface charges for a resistive long straight strip carrying a steady current. American Journal of Physics, 71:938–942, 2003.
(x ) ≡ dI 0 (x ) dx . Therefore the functions, where I 0 ' integral of Eq. (8) for z > 0 is given by (using the ν
∞
∞ J (x ρ a ) e − xn z I 0 (kρ ) sin (kz ) π dk = − π ∑ 0 n 2 k xn n =1 J 1 (x n )
∫ I (ka ) 0 0
a
(37 )
Analogously, the integral of Eq. (8) for z < 0 is given by: ∞
∞ J (x ρ a ) e xn z a I 0 (kρ ) sin (kz ) π dk = − + π ∑ 0 n 2 k xn n =1 J1 (x n )
∫ I (ka ) 0 0
(38)
[10] M.A. Heald. Electric fields and charges in elementary circuits. American Journal of Physics, 52:522–526, 1984. [11] J.D. Jackson. Surface charges on circuit wires and resistors play three roles. American Journal of Physics, 64:855–870, 1996. [12] O.D. Jefimenko. Electricity and Magnetism. Plenum, New York, Prob. 9.33 and Fig. 14.7, 1966.
Revista Facultad de Ingeniería, Chile, Vol. 12 Nº2, 2004
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J.A. Hernandes, E.Capelas de Oliveira, A.K.T. Assis
[13] O.D. Jefimenko. Electricity and Magnetism. Electret Scientific Company, Star City, 2nd edition, pp. 318 and 509-511, 1989.
[17] E. Butkov. Mathematical Physics. Wesley, Reading, pp. 403, 1968.
Addison-
[14] A. Sommerfeld. Electrodynamics. Academic Press, New York, pp. 125-130, 1964.
[18] O.D. Jefimenko. Electricity and Magnetism. Electret Scientific Company, Star City, 2nd edition, Prob9.33 and Fig. 14.7, 1989.
[15] J.A. Stratton. Electromagnetic Theory. McGrawHill, New York, pp. 262, 1941.
[19] E. Butkov. Mathematical Physics. Wesley, Reading, pp. 312, 1968.
[16] O.D. Jefimenko. Electricity and Magnetism. Electret Scientific Company, Star City, 2nd edition, pp. 509-511, 1989.
20
Revista Facultad de Ingeniería, Chile, Vol. 12 Nº2, 2004
Addison-
Hernandes, J. A.; Oliveira Capelas de, E.; Assis, A. K. T. Potential, electric field and surface charges close to the battery for a resistive cylindrical shell carrying a steady longitudinal current Revista Facultad de Ingeniería, vol. 12, núm. 2, 2004, pp. 13-20 Universidad de Tarapacá Arica, Chile
Available in: http://www.redalyc.org/articulo.oa?id=11412202
How to cite Complete issue More information about this article Journal's homepage in redalyc.org
Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative
REVISTA FACULTAD DE INGENIERÍA, U.T.A. (CHILE), VOL. 12 Nº2, 2004, pp. 13-20
POTENTIAL, ELECTRIC FIELD AND SURFACE CHARGES CLOSE TO THE BATTERY FOR A RESISTIVE CYLINDRICAL SHELL CARRYING A STEADY LONGITUDINAL CURRENT J. A. Hernandes1
E. Capelas de Oliveira2
A. K. T. Assis3
Recibido el 12 de mayo de 2004, aceptado el 30 de agosto de 2004 RESUMEN En este trabajo consideramos una capa resistiva cilíndrica que transporta una corriente constante. Una batería genera la corriente en el centro del conductor. Estudiamos el comportamiento del potencial, campo eléctrico y cargas superficiales cerca de la batería. Palabras clave: Potencial eléctrico, potencial cerca de la batería, cargas superficiales. ABSTRACT In this work we consider a long, resistive cylindrical shell carrying a steady current. A battery in the middle of the wire generates the current. We study the behavior of the potential, electric field and surface charges close to the battery. Keywords: Electric potential, potential close to the battery, surface charges. INTRODUCTION There has been recently a great interest in the electric field outside resistive conductors carrying steady currents and a number of problems have been published in the literature: coaxial cables, [14], [8], [13], [1] and [2]; cylindrical shell with azimuthal current, [12], [10] and [16]; planes, [13]; twin leads, [15] and [4]; a long strip, [9]; and a long straight cylindrical conductor with longitudinal current, [5]. In this last case it has only been considered the region far from the battery. It was then found that the potential and surface charges vary linearly with the longitudinal component. Here we analyse the situation close to the battery in order to understand the behaviour of surface charges at a discontinuity in the potential. We consider a hollow cylindrical shell of radius a and >> a . The shell has an uniform length with resistivity and carries a steady current I along the positive z direction which coincides with its axis. We suppose an idealized linear battery at (ρ ,ϕ , z ) = (a,ϕ ,0 ) , see Fig. 1, in analogy with Heald's treatment which considered a “line” battery at (ρ ,ϕ , z ) = (a,π , z ) driving current azimuthally in a uniform cylindrical resistive 1
sheet, [10]. We suppose a vacuum inside and outside the shell.
Fig. 1 A resistive cylindrical shell of radius a and length carrying a steady current in the z direction. There is a linear battery in z = 0 generating a difference of potential of 2φ 0 . Ohm's law can be
written as J = gE , where
J = (I / 2πat )zˆ is the volume current density ( t 0) = (φ R − φ L ) + R + φ0 2
(1) (2) Fig. 3
The potential of Fig. 2 can be decomposed in two parts: the shell held at constant but discontinuous potentials (dashed lines), and the shell with a continuous potential varying linearly with the longitudinal coordinate (continuous line).
ELECTROSTATIC SOLUTION OF THE CYLINDRICAL SHELL WITH IS TWO HALVES HELD AT CONSTANT AND OPPOSITE POTENTIALS
Fig. 2 Potential along the shell. Our goal is to find the potential and electric field at ρ < a and at ρ > a given the boundary conditions above. Then we can obtain the surface charge densities at the inner and outer surfaces of the shell. This problem can be separated in two parts: a) the electrostatic situation of a cylindrical shell separated at z = 0 by a thin insulating barrier held at −φ 0 for z < 0 and at φ 0 for z > 0 ; and b) a continuous linear potential
Suppose two semi-infinite cylindrical shells, located at z < 0 and z > 0 , see Fig. 1. The shell at z < 0 is held at the constant potential −φ 0 , while the shell at z > 0 is held at φ 0 , [17]. We can solve Laplace's equation,
∇ 2φ = 0 , in cylindrical coordinates using separation of variables in the form φ jk (ρ ,ϕ , z ) = R jk (ρ )Φ j (ϕ )Z k (z ) , where the functions R jk , Φ j and Z k obey the equations: d 2 R jk dρ
2
+
1 dR jk 2 j 2 − k + 2 R jk = 0 , ρ dρ ρ d 2Φ j
φ (a,ϕ , z ) = (φ R − φ L ) z + (φ R + φ L ) 2 for − 2 ≤ z ≤ 2 , see Fig. 3. The first part has been partially solved in [7] and the second one in [5]. Our intention is to deepen these studies in order to understand the behaviour close to the battery.
dϕ 2 d 2Zk dz 2
− j 2Φ j = 0 ,
(3)
+ k 2Zk = 0
The final solution φ (ρ ,ϕ , z ) is a linear combination of all possible solutions φ jk (ρ ,ϕ , z ) .
Due to the rotational symmetry of the system, the solution will not depend on ϕ . This means j = 0 and
Φ j = constant . 14
Revista Facultad de Ingeniería, Chile, Vol. 12 Nº2, 2004
Potential, electric field and surface charges close to the battery for a resistive cylindrical shell carrying a steady longitudinal current
In this situation, the boundary conditions are antisymmetric about the coordinate, z φ (a,ϕ ,− z ) = −φ (a,ϕ , z ) , so the solution must also be anti-symmetric at all points: φ (ρ ,ϕ ,− z ) = −φ (ρ ,ϕ , z ) and φ (ρ ,ϕ ,0 ) = 0 . Additionally, we must have a limited solution in both and coordinates, z ρ
φ (ρ ,ϕ , z → ∞ ) ≤ φ 0 < ∞
and
This leads to a solution for
φ (ρ → ∞,ϕ , z ) → 0 . Zk
of the form:
Z k (z ) = sin (kz ) . The equation for R jk is the modified
Bessel equation, whose solutions are I j (kρ ) (regular
In Eq. (9), xn are the zeroes of the Bessel function of order zero, J 0 (x n ) = 0 , with n = 1,2,
SOLUTION FOR THE CYLINDRICAL SHELL WITH STEADY CURRENT AND BATTERY Eqs. (13) and (14) of [5] give the solution of a
conducting wire with radius a , length >> a and carrying a steady current as: z φ + φL φ (ρ ≤ a,ϕ , z ) = (φ L − φ R ) + R 2
for ρ → 0 and irregular for ρ → ∞ ) and K j (kρ ) (regular for ρ → ∞ and irregular for ρ → 0 ), [6].
( ρ) z φ (ρ ≥ a, ϕ , z ) = (φ L − φ R ) + ( a) φR + φL ( ρ ) + ( a) 2
The general solution has then the form: ∞
φ (ρ ≤ a,ϕ , z ) = ∫ Ak I 0 (kρ )sin (kz )dk
(4)
0
∞
φ (ρ ≥ a,ϕ , z ) = ∫ Bk K 0 (kρ )sin (kz )dk
(5)
0
where the coefficients Ak and Bk must be determined from the boundary conditions. Eqs. (4) and (5) can be seen as a sine Fourier transform of a function Ψ : ∞
2 Ψ (ρ , ϕ , k ) = φ (ρ , ϕ , z ) (kz )dz = π 0
∫
π 2 Ak I 0 (kρ ), ρ ≤ a = π 2 Bk K 0 (kρ ) ρ ≥ a
(12)
The solution of the problem with the battery is the sum of Eqs. (8) and (11) for the region inside the shell, and the sum of Eqs. (10) and (12) for the region outside the shell, namely:
φ (ρ ≤ a, ϕ , z ) =
2φ 0 π
∞
∫ 0
I 0 (kρ ) (kz ) dk + I 0 (ka ) k
(13)
φ + φL + (φ L − φ R ) + R 2 z
(6)
∞ 2φ 0 K 0 (kρ ) φ (ρ ≥ a, ϕ , z ) = π 0 K 0 (ka ) ( ρ) z φR + φL + (φ L − φ R ) + ( a) 2
∫
Calculating Eq. (6) in ρ = a and applying the boundary conditions φ (a,ϕ , z < 0 ) = −φ 0 and φ (a,ϕ , z > 0 ) = φ 0 yields the coefficients Ak and Bk : 2φ 0 2φ 0 Ak = , Bk = πkI 0 (ka ) πkK 0 (ka )
(11)
(kz ) dk +
k ( ρ) ( a)
(14)
Fig. 4 shows the equipotentials with φ L = φ 0 , φ R = −φ 0
(7 )
and
a = 10 .
The final solution can be written as (see Appendix):
φ (ρ ≤ a,ϕ , z ) =
2φ 0 π
∞
I 0 (kρ ) sin (kz ) dk k
∫ 0 I 0 (ka )
−x z a z 1 ∞ J 0 (xn ρ a ) e n −∑ z 2 n=1 J1 (xn ) xn 2φ 0 ∞ K 0 (kρ ) sin (kz ) dk φ (ρ ≥ a,ϕ , z ) = ∫ π 0 K 0 (ka ) k
= 2φ 0
(8) (9)
(10)
Revista Facultad de Ingeniería, Chile, Vol. 12 Nº2, 2004
15
J.A. Hernandes, E.Capelas de Oliveira, A.K.T. Assis
internal and external densities of surface charge as given by, respectively (where ε 0 = 8.85 × 10 −12 C 2 N −1m −2 ):
(
)
σ 2 32 3 a − , ϕ , z = − lim ε 0 E ρ (ρ < a, ϕ , z ) = = =
2φ 0 ε 0 π
∞
ρ →a
I (ka )
∫ I 10 (ka) sin(kz )dk = 0
2φ 0 ε 0 z a z
(
∞
∑ e −x
n
(17 )
z a
n =1
)
σ 5786 a + , ϕ , z = lim ε 0 E ρ (ρ > a, ϕ , z ) = = Fig. 4
Equipotentials of conducting wire close to z = 0 . This is a plot of Eqs. (13) and (14) with φ L = −φ R = φ 0 and a = 10 . There is a steady current flowing along the positive z direction.
The electric field E can be obtained from the potential utilizing the relation E = −∇φ . This yields:
2φ E (ρ < a, ϕ , z ) = − 0 π ∞ ∞ ( ) I (kρ ) " # ρˆ 1 (kz )dk + zˆ I 0 kρ ! (kz )dk + I (ka ) 0 I 0 (ka ) 0 0 φ −φ (15) + zˆ L R
∫
2φ 0 ε 0 π
∞
ρ →a
K1 (ka )
∫ K 0 (ka ) sin(kz )dk +
(18)
0
ε (φ − φ ) ε (φ + φ ) − 04 L 4 R z + 0 R 4 L a ln ( a ) 2a ln ( a ) Fig. 5 shows these normalized densities of surface charge as a function of z a . Both of them diverge to infinity when z a > 1 the internal density of surface charge goes to zero faster than the external one.
∫
.
E (ρ > a, ϕ , z ) =
2φ 0 ⋅ π
∞ ∞ K (kρ ) )+*-, (kz )dk − zˆ K 0 (kρ ) $&%( ' (kz )dk + ⋅ ρˆ 1 K 0 (ka ) 0 K 0 (ka ) 0
∫
∫
− 1 ρˆ (φ L − φ R ) /z − φ R + φ L + 0 1 / 2 ( a) ρ + 0 1 / − φ φ ( ) + zˆ L / R 0 1 / ρ ( a)
Fig. 5
(16)
The surface charge distribution σ (a,ϕ , z ) can be found by applying Gauss' law and choosing a gaussian surface surrounding a small piece of the conductor surface. In the limit of an infinitesimal surface this yields the 16
Densities of surface charge as a function of the z (longitudinal) coordinate for a resistive hollow cylindrical shell carrying a steady current. The dashed (continuous) lines is the normalized internal (external) density of surface charges. Both densities are normalized by the internal density ( ρ = a − ) at z = a
Revista Facultad de Ingeniería, Chile, Vol. 12 Nº2, 2004
Potential, electric field and surface charges close to the battery for a resistive cylindrical shell carrying a steady longitudinal current
DISCUSSION Jefimenko and Heald considered a similar geometry (resistive cylindrical shell of radius a coaxial with the z axis) but carrying a steady azimuthal current, [18], [10] and [16]. A line battery located at (ρ ,ϕ , z ) = (a,π , z ) had its terminals at potentials ±φ 0 . They found equal internal and external surface charge densities given by:
σ ; 0 z a ln ( a ) charge distribution is null, σ (a, z 0 ) = 0 . In the interval A 1 ≤ a ≤ 100 , equating Eq. (31) to zero yields
A
z 0 a = 0.0890 + 0.05068 a ,
A
for z 0 ≈ 0.507 a >> 1 . Although our solution is valid only for
A
or
z 0 . We choose a contour of the type shown in Fig. 6 for z > 0 . The contour integral can thus be divided in three parts: along the real k axis, along the path Cr and along the path C R . The integral along the path Cr is given by:
lim
r →0
∫
Cr
I 0 (kρ ) e ikz dk = I 0 (ka ) k
( ) ( )ire ϕ dϕ = −iπ = lim ∫ ( e ϕ a) re ϕ π I r 0
r →0
I 0 r e iϕ ρ exp ikre iϕ 0
i
Revista Facultad de Ingeniería, Chile, Vol. 12 Nº2, 2004
i
i
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Potential, electric field and surface charges close to the battery for a resistive cylindrical shell carrying a steady longitudinal current
REFERENCES [1] A.K.T. Assis and J.I. Cisneros. The problem of surface charges and fields in coaxial cables and its importance for relativistic physics. Open Questions in Relativistic Physics, pp. 177–185. F. Selleri (editor), Apeiron, Montreal, pp.508-511, 1998.
Fig. 6
Contour to calculate the integral of Eq. (8), for z > 0 . For z < 0 we choose a symmetrical contour reflected at the horizontal (real k ) axis
The integral along the path C R is limited, and vanishes for R → ∞ : I (kρ ) e ikz
e ikz
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0 dk ≤ lim ∫ dk = 0 ∫ R →∞ C I 0 (ka ) k R→∞ C k R R
lim
Using Cauchy's theorem, the integral I given by:
lim
∫
r →0 R →∞ C
I 0 (kρ ) e ikz dk = I 0 (ka ) k
∞
∫
−∞
I 0 (kρ ) e ikz dk − iπ = I 0 (ka ) k
I 0 (k n ρ ) e ikn z = 2iπ Res(k n ) = 2iπ I (k a ) ak n n =1 n =1 1 n ∞
∞
∑
∑
is
[2] A.K.T. Assis and J.I. Cisneros. Surface charges and fields in a resistive coaxial cable carrying a constant current. IEEE Transactions on Circuits and Systems I, 47:63–66, 2000. [3] A.K.T. Assis, J.A. Hernandes and J.E. Lamesa. Surface charges in conductor plates carrying constant currents. Foundations of Physics, 31:1501–1511, 2001. [4] A.K.T. Assis and A.J. Mania. Surface charges and electric field in a two-wire resistive transmission line. Revista Brasileira de Ensino de Física, 21:469–475, 1999. [5] A.K.T. Assis, W.A. Rodrigues Jr., and A.J. Mania. The electric field outside a stationary resistive wire carrying a constant current. Foundations of Physics, 29:729–753, 1999.
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where Res(k n ) is the residue of the integrand in
[6] G.B. Arfken and H.J. Weber. Mathematical Methods for Physicists. Academic Press, San Diego, 4th edition, Chap. 11, 1995. [7] E. Butkov. Mathematical Physics. Wesley, Reading, pp. 403, 1968.
Addison-
k = k n = ixn a . In the equation above we utilized the (x ) = I1 (x ) involving modified Bessel relation I 0 '
[8] D.J. Griffiths. Introduction to Electrodynamics. Prentice Hall, New Jersey, 3rd edition, pp. 336337, 1999.
relation Jν (ix ) = i Iν (x ) ):
[9] J.A. Hernandes and A.K.T. Assis. The potential, electric field and surface charges for a resistive long straight strip carrying a steady current. American Journal of Physics, 71:938–942, 2003.
(x ) ≡ dI 0 (x ) dx . Therefore the functions, where I 0 ' integral of Eq. (8) for z > 0 is given by (using the ν
∞
∞ J (x ρ a ) e − xn z I 0 (kρ ) sin (kz ) π dk = − π ∑ 0 n 2 k xn n =1 J 1 (x n )
∫ I (ka ) 0 0
a
(37 )
Analogously, the integral of Eq. (8) for z < 0 is given by: ∞
∞ J (x ρ a ) e xn z a I 0 (kρ ) sin (kz ) π dk = − + π ∑ 0 n 2 k xn n =1 J1 (x n )
∫ I (ka ) 0 0
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[10] M.A. Heald. Electric fields and charges in elementary circuits. American Journal of Physics, 52:522–526, 1984. [11] J.D. Jackson. Surface charges on circuit wires and resistors play three roles. American Journal of Physics, 64:855–870, 1996. [12] O.D. Jefimenko. Electricity and Magnetism. Plenum, New York, Prob. 9.33 and Fig. 14.7, 1966.
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J.A. Hernandes, E.Capelas de Oliveira, A.K.T. Assis
[13] O.D. Jefimenko. Electricity and Magnetism. Electret Scientific Company, Star City, 2nd edition, pp. 318 and 509-511, 1989.
[17] E. Butkov. Mathematical Physics. Wesley, Reading, pp. 403, 1968.
Addison-
[14] A. Sommerfeld. Electrodynamics. Academic Press, New York, pp. 125-130, 1964.
[18] O.D. Jefimenko. Electricity and Magnetism. Electret Scientific Company, Star City, 2nd edition, Prob9.33 and Fig. 14.7, 1989.
[15] J.A. Stratton. Electromagnetic Theory. McGrawHill, New York, pp. 262, 1941.
[19] E. Butkov. Mathematical Physics. Wesley, Reading, pp. 312, 1968.
[16] O.D. Jefimenko. Electricity and Magnetism. Electret Scientific Company, Star City, 2nd edition, pp. 509-511, 1989.
20
Revista Facultad de Ingeniería, Chile, Vol. 12 Nº2, 2004
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