On the vector solutions of Maxwell equations

in spherical coordinate systems

Swarnav
E.A. Majumder,
Matute
student,
Department of Physics,
AJC Bose College
[email protected]
arXiv:physics/0512261v1 [physics.class-ph] 29 Dec 2005

The Maxwell equations for the spherical components of the electromagnetic fields outside sources
do not separate into equations for each component alone. We show, however, that general solutions
can be obtained by separation of variables in the case of azimuthal symmetry. Boundary condi-
tions are easier to apply to these solutions, and their forms highlight the similarities and differences
between the electric and magnetic cases in both time-independent and time-dependent situations.
Instructive examples of direct calculation of electric and magnetic fields from localized charge and
current distributions are presented.
Keywords: Maxwell equations; spherical coordinates; electric and magnetic fields; boundary-value
problems.

Las ecuaciones de Maxwell para las componentes esféricas d

libres de fuentes no son separables en ecuaciones para cada u
sin embargo, que soluciones generales pueden ser obtenidas
de simetrı́a azimutal. Las condiciones de borde son fácile
formas destacan las similitudes y diferencias entre los cas
situaciones independientes del tiempo como para las de tiem
instructivos de cálculos directos de campos eléctricos y
localizadas de cargas y corrientes.
Descriptores:
mas con condiciones de borde.

PACS: 03.50.De; 41.20.Cv; 41.20.Gz

1. INTRODUCTION drawn from the second into the first region. The interior
and exterior fields satisfy the homogeneous vector wave
The Maxwell equations for the electromagnetic field equations
vectors, expressed in the International System of Units ∂2E
(SI), are [1] ∇2 E − ǫµ = 0,
∂t2
∂B
∇ · D = ρ, ∇×E = − , ∂2B
∂t ∇2 B − ǫµ = 0, (4)
∂t2
∂D
∇ · B = 0, ∇×H = J + , (1) which are obtained from Eqs. (1) and (2) for regions
∂t free of charge and current by combining the two curl
where the source terms ρ and J describe the densities equations and making use of the divergence equations
of electric charge and current, respectively. For a linear, together with the vector identity
isotropic medium D and H are connected with the basic
fields E and B by the constitutive relations ∇2 ( ) = ∇(∇· ) − ∇×(∇× ). (5)

D = ǫE, H = B/µ, (2) Changes in the electromagnetic fields propagate with

√
speed v = 1/ ǫµ .
where ǫ and µ are the permittivity and permeability of Without any loss of generality, we may consider only
the medium, respectively. harmonic time dependence for sources and fields:
The boundary conditions for fields at a boundary sur-
face between two different media are [2] ρ(r, t) = ρ(r)e−iωt , J(r, t) = J(r)e−iωt ,
n · (D1 − D2 ) = ρS , n×(E1 − E2 ) = 0,
E(r, t) = E(r)e−iωt , B(r, t) = B(r)e−iωt , (6)
n · (B1 − B2 ) = 0, n×(H1 − H2 ) = JS , (3) where the real part of each expression is implied. Equa-
tion (4) then becomes time-independent:
where ρS and JS denote the surface charge and current
densities, respectively, and the normal unit vector n is ∇2 E + k 2 E = 0, ∇2 B + k 2 B = 0, (7)
2

where k 2 = ǫµω 2 . These are vector Helmholtz equations The purpose of this paper is to get general solutions
for transverse fields having zero divergence. Their solu- of the electromagnetic vector equations in spherical co-
tions subject to arbitrary boundary conditions are con- ordinates with azimuthal symmetry using separation of
sidered more complicated than those of the correspond- variables in spite of having equations that mix field com-
ing scalar equations, since only in Cartesian coordinates ponents. Boundary conditions are easier to apply to
the Laplacian of a vector field is the vector sum of the these solutions, and their forms highlight the similari-
Laplacian of its separated components. For spherical co- ties and differences between the electric and magnetic
ordinates, as for any other curvilinear coordinate system, cases in both time-independent and time-dependent sit-
we are faced with a highly complicated set of three si- uations. The approach shows that boundary-value prob-
multaneous equations, each equation involving all three lems can be solved for the electric and magnetic vector
components of the vector field. This complication is well fields directly, and that the process involves the same
known and general techniques for solving these equa- kind of mathematics as the usual approach of solving for
tions have been developed, based on a dyadic Green’s potentials. The material in this work may be used in a
function which transforms the boundary conditions and beginning graduate course in classical electromagnetism
source densities into the vector solution [3]. We shall or mathematical methods for physicists. It is organized
show, however, that in the case of spherical boundary as follows. In Sec. 2 we describe the method for the
conditions with azimuthal symmetry, the solution can be static case showing how the mathematical complications
obtained more conveniently by means of separation of of solving the vector field equations are easily overcome
variables. Several applications of physical interest can by means of separation of variables. In Sec. 3 we ex-
then be treated in this simplified way, so avoiding the tend the method to discuss the case of time-varying fields.
dyadic method [4]. Concluding remarks are given in Sec. 4.
Actually, the usual technique for solving boundary-
value problems introduces the electromagnetic potentials
as intermediary field quantities. These are defined by [5] 2. STATIC FIELDS

∂A For steady-state electric and magnetic phenomena, the

B = ∇×A, E = −∇φ − , (8)
∂t fields outside sources satisfy the vector Laplace equations
with the subsidiary Lorentz condition
∇2 E = 0, ∇2 B = 0, (13)
∂φ
∇ · A + ǫµ = 0. (9) where only transverse components with zero divergence
∂t are involved. Supposing all the charge and current are on
It is then found that these potentials satisfy the inhomo- the bounding surfaces, solutions in different regions can
geneous wave equations be connected through the boundary conditions indicated
in Eq. (3). To demonstrate the features of the treat-
∂2φ ρ ment, we first consider boundary-value problems with
∇2 φ − ǫµ =− ,
∂t 2 ǫ azimuthal symmetry in electrostatics. The solution of
stationary current problems in magnetostatics is mathe-
∂2A matically identical.
∇2 A − ǫµ = −µJ, (10)
∂t2 Combining the expressions for ∇×(∇×E) = 0 and
∇ · E = 0 in spherical coordinates and assuming no ϕ-
which together with the Lorentz condition form a set
dependence, we find using Eq. (5) that the components
of equations equivalent to the Maxwell equations. The
of the electric field Er and Eθ satisfy the equations
boundary conditions for the potentials may be deduced
from Eq. (3). 1 ∂2 2
For fields that vary with an angular frequency ω, i.e. (∇2 E)r = (r Er )
r2 ∂r2
1 ∂ ∂Er
φ(r, t) = φ(r)e−iωt , A(r, t) = A(r)e−iωt , (11) + 2 (sin θ ) = 0, (14)
r sin θ ∂θ ∂θ
we get equations that do not depend on time in regions
free of charge and current: 1 ∂2 1 ∂ 2 Er
(∇2 E)θ = (rEθ ) − = 0. (15)
r ∂r2 r ∂r∂θ
2 2
∇ φ + k φ = 0,
Equation (14) is for Er alone, whereas Eq. (15) involves
both components. There is also a separated equation for
∇2 A + k 2 A = 0, (12) Eϕ :

which are like those in Eq. (7) for the electric and mag- 1 ∂2 1 ∂
(∇2 E)ϕ = (rEϕ ) + 2
netic induction fields, so that in general we also confront, r ∂r 2 r sin θ ∂θ
for the vector potential, the mathematical complexities ∂Eϕ 1
mentioned above for the electromagnetic fields. × (sin θ ) − 2 2 Eϕ = 0. (16)
∂θ r sin θ
3

In this paper, however, we will not be concerned about for n ≥ 1 with ao = 0; this null factor in Eq. (23) means
those cylindrical symmetry cases where only the ϕ- the absence of static field terms of the 1/r type, which
component of the vector field is nonzero because a scalar are in reality typical of radiative fields as shown below.
technique of separation of variables is already known to Clearly, the solutions given in Eqs. (23), (24) and (25)
obtain the solution [6]. satisfy Eq. (18). The coefficients an and bn are to be de-
Using the transverse condition termined from the boundary conditions. For complete-
ness, we include here the well-behaved general solution
1 ∂ 2 1 ∂ of Eq. (16):
∇·E= 2
(r Er ) + (sin θ Eθ ) = 0, (17)
r ∂r r sin θ ∂θ
∞
where azimuthal symmetry is assumed, Eq. (14) implies
X
n dn d
Eϕ (r, θ) = cn r + n+1 Pn (cos θ). (26)
n=0
r dθ
∂ ∂Er
(rEθ ) − = 0, (18)
∂r ∂θ Thus, Eqs. (23)-(26) formally give all three components
which is consistent with Eq. (15). Thus, to obtain Eθ of the electric field. The same type of equations applies
from Er we can consider either Eq. (17) or Eq. (18). in magnetostatics. However, the boundary conditions of
These equations correspond to choosing a gauge when Eq. (3) will make the difference, implying in particular
this method is applied to the vector potential. that b◦ = 0 in the series expansion of Eq. (23) in magne-
Now, in order to solve Eq. (14) for Er , we refer to the tostatics; this being primarily related to the absence of
method of separation of variables and write the product magnetic monopoles.
form To illustrate the use of the above formulas, we consider
the simple example of the electric field due to a ring of
u(r) radius a with total charge Q uniformly distributed and
Er (r, θ) = P (θ), (19)
r2 lying in the x-y plane. It is usually solved through the
scalar potential method by using the result of the po-
which leads to the following separated differential equa-
tential along the z-axis obtained from Coulomb’s law [7].
tions:
The surface charge density on r = a, localized at θ = π/2,
d2 u n(n + 1) is written as
− u = 0, (20)
dr2 r2 Q
ρS (a, θ) = δ(cos θ), (27)
1 d dP 2πa2
(sin θ ) + n(n + 1)P = 0, (21)
sin θ dθ dθ
which may be expanded using the well-known Legendre
where n(n + 1) is the separation constant. The solution series
of Eq. (20) is ∞
X 2n + 1
b δ(cos θ) = Pn (0) Pn (cos θ), (28)
u(r) = a rn+1 + , (22) n=0
2
rn
where a and b are arbitrary constants. Equation (21) with Pn (0) given by
is the Legendre equation of order n and the only solu-
tion which is single valued, finite and continuous over the (−1)n (2n + 1)!
P2n+1 (0) = 0, P2n (0) = . (29)
whole interval corresponds to the Legendre polynomial 22n (n!)2
Pn (cos θ), n being restricted to positive integer values.
Thus the general solution for Er is Taking into account the cylindrical symmetry of the sys-
tem, and the requirement that the series solutions in
∞
X bn Eqs. (23)-(25) have to be finite at the origin, vanish at
Er (r, θ) = an rn−1 + n+2 Pn (cos θ). (23) infinity and satisfy the boundary conditions of Eq. (3)
n=0
r
at r = a for all values of the angle θ, namely, Eθ con-
The simplest way of solving Eq. (17) for Eθ is to use the tinuous at r = a and Er discontinuous at r = a, it is
series expansion straightforwardly found that the spherical components
of the electric field are
∞
X d
Eθ (r, θ) = vn (r) Pn (cos θ), (24) Q X
∞

n=0
dθ Er (r, θ) = Pn (0) Pn (cos θ)
4πǫ◦ r2 n=0
where vn (r) are functions to be determined. By replacing  a n
Eqs. (23) and (24) into Eq. (17), it is found that
 (n + 1) , r>a
r


× (30)
an n−1 bn 1  −n r
 n+1
vn (r) = r − (25) , r<a

n n+1 r n+2
a
4

∞
Q X
∞ µ◦ Ia2 X 1
Eθ (r, θ) = − Pn (0) Pn1 (cos θ) Bθ (r, θ) = P (0) Pn1 (cos θ)
4πǫ◦ r2 n=0 2r3 n=0 n

 a n 1 a n−1
 , r>a


 n+1 r , r>a
 r
 
× (31) × (36)
 r
 n+1 
 1 r n+2
, r<a − , r<a
 
a n a

and Eϕ = 0, where Pn1 (cos θ) = (d/dθ) Pn (cos θ) is an Note that, as anticipated for magnetostatic problems, the
associated Legendre function. Note in particular that the coefficient b◦ in Eq. (23) is equal to zero. Also, as ex-
coefficient b◦ in Eq. (23) becomes Q/4πǫ◦ for r > a, as pected, the discontinuity of the nth component of Bθ in
expected. Also, the discontinuity of the nth component Eq. (36) at r = a is connected according to Eq. (3) with
of Er in Eq. (30) at r = a is connected according to the corresponding component of the surface current den-
Eq. (3) with the corresponding component of the surface sity JSϕ obtained from Eqs. (32) and (33). Another char-
charge density ρS obtained from Eqs. (27) and (28), ex- acteristic difference with the electrostatic analog is that
hibiting the unity of the multipole expansions of fields the coefficient Pn1 (0) appears instead of Pn (0). This can
and sources (see Ref. [8]). be traced to the fact that the inhomogeneous boundary
condition, as given by Eq. (3), is applied to the angular
To clarify the application of the formulas in the case of
component of the magnetic induction field in Eqs. (24)-
magnetostatics and also compare with electrostatics, we
(25), as opposed to the corresponding inhomogeneous
consider next the magnetic analog of the above example,
boundary condition acting on the radial component of
that is, the magnetic induction field from a circular cur-
the electric field in Eq. (23). The fields in Eqs. (35)-
rent loop of radius a lying in the x-y plane and carrying a
(36) are usually obtained through the vector potential
constant current I. The surface current density on r = a
method by using the expression of the magnetic induc-
can be written as
tion field along the z-axis calculated from the Biot and
I Savart law [6]. An alternative technique is mere integra-
JS (a, θ, ϕ) = δ(cos θ) ϕ̂, (32) tion of the vector potential [9]. Our treatment has the
a
advantage of introducing a considerable simplification on
where for the delta function is now convenient to use the the procedure of applying the boundary conditions on
expansion the magnetic induction field directly.

∞
X 2n + 1
δ(cos θ) = P 1 (0) Pn1 (cos θ), (33) 3. TIME-VARYING FIELDS
n=0
2n(n + 1) n
By using Eqs. (1), (2) and (6) it is seen that outside
which follows from the completeness relation for the sources the fields are related by
spherical harmonics after multiplication by e−iϕ and in-
tegration over ϕ. The values for Pn1 (0) are iω
E= ∇×B, (37)
k2
1 1 (−1)n+1 (2n + 1)!
P2n (0) = 0, P2n+1 (0) = . (34) so that we only need to solve Eq. (7) for B. Alternatively,
22n (n!)2 we can solve for E, and obtain B through the expression
Because of the cylindrical symmetry of the system, Bϕ = i
0. By requiring that the field be finite at the origin, B=− ∇×E. (38)
ω
vanish at infinity and satisfy the boundary conditions of
Eq. (3) at r = a, the series solutions in Eqs. (23)-(25) In the following, we choose to deal with the Helmholtz
for the magnetic case lead to the following radial and equation for the magnetic induction field. The reason is
angular components of the magnetic induction field: to exhibit similarities and differences with the static case
treated in Sec. 2.
∞ In the case of spherical boundary surfaces with az-
µ◦ Ia2 X 1
Br (r, θ) = − P (0) Pn (cos θ) imuthal symmetry, the Br and Bθ components of the
2r3 n=0 n magnetic induction satisfy the following equations:
 a n−1
 , r>a 1 ∂2 2 1
 r (∇2 B)r + k 2 Br = (r Br ) + 2

2
r ∂r 2 r sin θ
× (35)
 r n+2 , r < a ∂ ∂Br

) + k 2 Br = 0, (39)

× (sin θ
a ∂θ ∂θ
5

1 ∂2 1 ∂ 2 Br the time-varying version of the case solved in Sec. 2. The

(∇2 B)θ + k 2 Bθ = 2
(rBθ ) −
r ∂r r ∂r∂θ surface density current on r = a is then
+ k 2 Bθ = 0. (40)
I◦
JS (a, θ, ϕ, t) = δ(cos θ) e−iωt ϕ̂, (48)
Similarly, for the Bϕ component we would have the equa- a
tion
which can be expanded using Eq. (33). The complete se-
1 ∂2 1 ∂ ries solution of the Helmholtz equation for the magnetic
(∇2 B)ϕ + k 2 Bϕ = 2
(rBϕ ) + 2 induction field, which is finite at the origin, represents
r ∂r r sin θ ∂θ
∂Bϕ 1 outgoing waves at infinity and satisfies the boundary con-
× (sin θ ) − 2 2 Bϕ + k 2 Bϕ = 0. (41) ditions of Eq. (3) at r = a, becomes
∂θ r sin θ
∞
These are analogous to Eqs. (14), (15) and (16) in con- µ◦ I◦ ka −iωt X
Br (r, θ, t) = −i e (2n + 1)Pn1 (0)
nection with the vector Laplace equation. In order to 2r n=0
solve Eq. (39) we let
 jn (ka) h(1)

n (kr)
j(r) ×Pn (cos θ) (49)
Br (r, θ) = P (θ), (42) 
jn (kr) h(1) (ka)
r n

whereupon separation yields

∞
µ◦ I◦ k 2 a −iωt X 2n + 1 1
d2 j 2 dj

n(n + 1)
Bθ (r, θ, t) = −i e P (0)
+ + k −2
j = 0, (43) 2 n=0
n(n + 1) n
dr2 r dr r2  h n (1) i
(1)
 jn (ka) hn−1 (kr) − kr hn (kr)


and Eq. (21), where the constant n(n + 1) is the sepa-
×Pn1 (cos θ) i (50)
ration parameter. Equation (43) is the spherical Bessel  (1)
 h n
equation of order n with variable kr. Therefore, the gen-
 hn (ka) jn−1 (kr) − jn (kr)
kr
eral solution for Br is
and Bϕ = 0, where the upper line holds for r > a and
∞
the lower line for r < a. As noted above, the coefficient

X jn (kr) nn (kr)
Br (r, θ) = an + bn Pn (cos θ). (44) a◦ in Eq. (44) indeed vanishes. Also, the discontinuity
n=0
r r
of the nth component of Bθ in Eq. (50) at r = a is con-
Depending on boundary conditions, the spherical Hankel nected, according to Eq. (3), with the nth component of
(1,2) the surface current density JSϕ obtained from Eqs. (48)
functions hn instead of the spherical Bessel functions and (33). A characteristic difference between this time-
jn , nn may be used. For Bθ we again write
varying problem and the corresponding static case is the
∞ appearance of the spherical Bessel functions, which are
X d solutions of the radial part of the Helmholtz equation in
Bθ (r, θ) = wn (r) Pn (cos θ), (45)
n=0
dθ spherical coordinates. Using their limiting values [10], it
can be seen that for k → 0 the static results in Eqs. (35)
and use ∇ · B = 0 to obtain now and (36) are obtained, as mathematically and physically
expected. On the other hand, the radiative part of the
an d
wn = [r jn (kr)] external magnetic induction field, which decreases as 1/r,
n(n + 1)r dr is given by
bn d
+ [r nn (kr)], (46) ∞
n(n + 1)r dr µ◦ I◦ ka i(kr−ωt) X (4n + 3)(2n − 1)!
B(r, θ, t) = θ̂ e
4r n=0
22n n!(n + 1)!
for n ≥ 1 with a◦ = b◦ = 0. The other coefficients an
1
and bn are determined so that the boundary conditions ×j2n+1 (ka) P2n+1 (cos θ). (51)
for the vector field are exactly satisfied. In the case of
the Bϕ component, the general solution is In the dipole approximation, ka ≪ 1, this becomes the
radiative magnetic induction field from an oscillating
∞
X d magnetic dipole of magnetic moment m = πa2 I◦ ẑ:
Bϕ (r, θ) = [cn jn (kr) + dn nn (kr)] Pn (cos θ). (47)
n=0
dθ µ◦ k 2 ei(kr−ωt)
B(r, t) = (r̂×m)×r̂ . (52)
4π r
The same type of equations applies for the electric field.
As an example, we shall consider the problem of the The magnetic induction field in Eqs. (49) and (50) can be
magnetic induction field from a current I = I◦ e−iωt in seen to be just that which is obtained with the more ar-
a circular loop of radius a lying in the x-y plane. It is duous technique of a dyadic Green’s function expanded in
6

vector spherical harmonics and applied to the vector po- illustrated the use of these formulas with direct calcula-
tential, which, by symmetry, only has the ϕ-component tions of electric and magnetic induction fields from local-
different from zero [3]. As we have shown, a direct ized charge and current distributions, without involving
calculation of the electromagnetic field with r- and θ- the electromagnetic potentials. Boundary conditions are
components is much simplified if separation of variables easier to apply to these solutions, and their forms high-
is used. light the similarities and differences between the electric
and magnetic cases in both time-independent and time-
dependent situations. Finally, we remark that in cylin-
4. CONCLUSION drical coordinates, the other commonly used curvilinear
coordinate system, the Maxwell equations do separate
For spherical coordinate systems, the Maxwell equa- into equations for each vector component alone if there
tions outside sources lead to coupled equations involving is cylindrical symmetry, so that the method of separation
all three components of the electromagnetic fields. In of variables can be used directly.
general, the statement is that one cannot separate spher-
ical components of the Maxwell equations, and extensive
techniques for solving the vector equations have been de-
Acknowledgments
veloped which introduce vector spherical harmonics and
use dyadic methods. We have shown, however, that sepa-
ration of variables is still possible in the case of azimuthal This work was partially supported by the Departa-
symmetry, and so general solutions for each component of mento de Investigaciones Cientı́ficas y Tecnológicas, Uni
the electromagnetic vector fields were obtained. We have

[1] W.K.H. Panofsky and M. Phillips, Classical Electric- [6] G. Arfken and H. Weber, Mathematical Methods for
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sachusetts, 1962), 2nd ed., Chap. 9. Chap. 12.
[2] Reference [1], Chap. 13. [7] Reference [5], Chap. 3.
[3] P.M. Morse and H. Feshbach, Methods of Theoreti- [8] E. Ley-Koo and A. Góngora-T, Rev. Mex. Fı́s. 34 (1988)
cal Physics (McGraw-Hill, New York, 1953), Vol. 2, 645.
Chap. 13. [9] Reference [5], Chap. 5.
[4] E.A. Matute, Am. J. Phys. 67 (1999) 786. [10] Reference [6], Chap. 11.
[5] J.D. Jackson, Classical Electrodynamics (Wiley, New
York, 1998), 3rd ed., Chap. 6.