Spherical volume averages of static electric and magnetic fields using Coulomb and Biot-Savart laws Ben Yu-Kuang Hu∗
arXiv:0902.0037v1 [physics.class-ph] 31 Jan 2009
Department of Physics, University of Akron, Akron, OH 44325-4001 (Dated: January 31, 2009)
Abstract We present derivations of the expressions for the spherical volume averages of static electric and magnetic fields that are virtually identical. These derivations utilize the Coulomb and Biot-Savart laws, and make no use of vector calculus identities or potentials.
1
I.
INTRODUCTION
The average of static electric or magnetic fields over a sphere has been used to obtain important results, such as the macroscopic electric field inside a dielectric1 and the presence of δ-function electric (magnetic) fields at the position of electric (magnetic) point dipoles.2 In both electric and magnetic cases, if the sources of the fields are outside the averaging sphere, the average field equals the value of the field at the center of the sphere, whereas for sources inside the averaging sphere, the average electric and magnetic fields are proportional to the electric and magnetic dipole moments of the sources, respectively. Textbooks typically only treat the electric field case.3 Only a handful of authors2,4 treat both electric and magnetic cases, probably because the magnetic field case is considered to be “tough” by undergraduate standards,5 since the derivations typically employ the magnetic vector potential and vector calculus identities. This paper presents derivations of both the electric and magnetic field cases that are virtually identical and are elementary, in the sense that they rely on Coulomb’s and the Biot-Savart laws and make no use of vector calculus identities or potentials. Coulomb’s law states that the electric field for a unit point charge is G(r) = k
r , r3
(1)
where r is a displacement vector from the point charge, r = |r | and k = (4πǫ0 )−1 and 1 in SI and Gaussian units, respectively. In terms of G, the electric field the electric field E at point s due to a static charge distribution ρ(r ′ ) is Z Z ′ ′ ′ E(s) = dr ρ(r ) G(s − r ) = − dr ′ ρ(r ′ ) G(r ′ − s).
(2)
Define Gr (r ′ ) to be the average of G over a spherical surface of radius r around a point r ′ , as shown in Fig. 1. An apparently little-known result that is critical in following discussion is (see the appendix for derivations) Gr (r ′ ) = k
r′ Θ(r ′ − r) r ′3
(3)
where Θ is the Heavyside step function.6 Eq. (3) implies that the average of the electric field over the surface of the sphere for a point charge outside the sphere (r ′ > r) equals the electric field at the center of the sphere, whereas a point charge inside a sphere (r ′ < r) 2
contributes zero to the average of the electric field over the surface of the sphere. In a sense, this result is the antithesis of the integral form of Gauss’ law,7 but it is not as general as Gauss’ law because it applies only to spherical surfaces.
II.
VOLUME AVERAGES OF STATIC ELECTRIC FIELDS
Let hEiR to be average of the electric field E about a spherical volume of radius R and Er (0) be the average electric field over a spherical shell of radius r, both centered around the origin. Then, hEiR
3 ≡ 4πR3
Z
R
3 dr E(r) = 3 R r=0
Z
R
dr r 2 Er (0).
(4)
0
For a given charge distribution ρ(r ′ ), Er (0) can be obtained by setting s = 0 and averaging both sides of Eq. (2) over a spherical shell of radius r, yielding Z Er (0) = − dr ′ ρ(r ′ ) Gr (r ′ ). A.
(5)
Single point charge
We first examine the case of a single point charge. Because electric fields obey the principle of linear superposition, what holds for a single point charge is generalizable to cases of many charges and/or continuous charge distributions. Substituting the charge distribution for a point charge q at d, ρ(r ′ ) = q δ(r ′ −d), into Eq. (5) gives Er (0) = −qGr (d). Substituting this into Eq. (4) and using Eq. (3) gives 3kqd hEiR = − 3 3 dR
Z
R 0
dr r 2 Θ(d − r).
(6)
If the point charge is outside the averaging sphere of radius R, then d > R and the integral (out)
in Eq. (6) equals R3 /3. Hence, hEiR
= −kqd/d3 = E(0), the electric field at the center
of the sphere due to the point charge. On the other hand, if the point charge is inside the averaging sphere, then d < R and the integral in Eq. (6) equals d3 /3, in which case (in)
hEiR = −kp/R3 , where p = qd is the dipole moment of the point charge relative to the center of the sphere.
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B.
Arbitrary charge distribution
We now obtain these results more rigorously for an arbitrary charge distribution ρ(r ′ ). Substituting Eqs. (3) and (5) into Eq. (4) yields, Z ∞ Z R ′ ′ 3 ′ k ρ(r ) r 2 hEiR = − 3 dr . dr r R 0 r ′3 r ′ =r
(7)
where the effect of the Θ function in Eq. (3) is to restrict the r ′ integration to the region |r ′ | > r. We now consider separately the contribution of the charges outside and inside the sphere.
1.
Sources outside the sphere
The contribution due to charges outside the sphere of radius R corresponds to
R∞
r ′ =R
dr ′
in Eq. (7), which makes the term in square parentheses independent of r. The integration over r gives R3 /3, yielding (out) hEiR
=−
Z
∞
dr ′
r ′ =R
k ρ(r ′ ) r ′ = E (out) (0) r ′3 (out)
the electric field at the origin. Therefore, in general, hEiR
(8)
= E (out) at the center of the
sphere.
2.
Sources inside the sphere
Sources inside the sphere of radius R correspond to |r ′ | < R in Eq. (7). This gives (in) hEiR
Z R Z R ′ ′ 3 2 ′ k ρ(r ) r =− 3 dr r dr R 0 r ′3 r ′ =r Z R Z ′ ρ(r ′ ) r ′ r 3k dr ′ dr r 2 =− 3 R r′ =0 r ′3 0 Z R k kp =− 3 dr ′ ρ(r ′ ) r ′ = − 3 , R r′ =0 R
(9)
R where p = dr ′ ρ(r ′ ) r ′ is the dipole moment relative to the center of the sphere, and we RR RR RR R r′ have used 0 dr r dr ′ = 0 dr ′ 0 dr, as shown in Fig. 2.
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III.
MAGNETIC FIELD CASE
A static magnetic field B is related to the charge current density J by the Biot-Savart law, B(s) =
Z
dr ′ J (r ′ ) × G(s − r ′ ),
(10)
where k = µ0 /(4π) or c−1 for SI or gaussian, respectively. Comparison of Eq. (10) with Eq. (2) shows that the derivation for the magnetic field case can be copied wholesale from the electric field case by substituting E → B and ρ → J × at every step. (The only point of caution is that the cross product anti-commutes, so care must be taken to preserve the order of J and G or r ′ .) Using these substitutions in Eq. (8) gives, for current sources outside the averaging sphere, (out) hBiR
=−
Z
∞
dr ′ r ′ =R
k J (r ′ ) × r ′ = B(0), r ′3
(11)
the magnetic field at the center of the sphere. For current sources inside the averaging sphere, using the substitutions in Eq. (9) gives (in) hBiR
where m =
1 2
R
k =− 3 R
Z
R
r ′ =0
dr ′ J (r ′ ) × r ′ =
2km , R3
(12)
dr ′ r ′ × J (r ′ ) is the magnetic dipole moment.8
APPENDIX A: DERIVATION OF EQ. (3)
By definition, 1 Gr (r ) = 4πr 2 ′
Z
dA′′ G(r ′′ ),
(A1)
ˆ where the integral is over the spherical shell of radius r around r ′ . Choose r ′ = r ′ z, ˆ = Gr,z zˆ = where zˆ is the unit vector in the z-direction. By azimuthal symmetry, Gr (r ′ z) R ˆ where Gz is the z-component of G. By dividing the spherical surface (4πr 2 )−1 dA′′ Gz (r ′′ )z, Rπ into strips dA′′ = r 2 sin θ dθ (see Fig. 3), Gr,z = 12 0 dθ sin θ Gz (r ′′ ). Letting µ = cos θ, and
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using Gz (r ′′ ) = krz′′ /|r ′′|3 gives Z rµ + r ′ k 1 dµ 2 Gr,z = 2 −1 (r + r ′2 + 2r ′ rµ)3/2 Z 1 k ∂ 1 =− dµ 2 2 ∂r ′ −1 (r + r ′2 + 2r ′ rµ)1/2 #1 "p r 2 + r ′2 + 2r ′ rµ k ∂ k ∂ |r + r ′ | − |r − r ′ | =− =− 2 ∂r ′ r′r 2 ∂r ′ r′r µ=−1 ∂r −1 =0 for r ′ < r; −k ∂r ′ = ′−1 k −k ∂r = for r ′ > r, ∂r ′ r ′2
(A2)
which implies Eq. (3).
This result can also be obtained by using the well known property that the average of an electrostatic potential of a point charge Q over the surface of a sphere of radius r equals the potential at the center if the charge is outside the sphere, and Q/(4πǫ0 r) if the charge is inside.9 Stated mathematically, if V (r ′′ ) = k/|r ′′ |, then the average of V over a spherical
shell of radius r around r ′ is
Vr (r ′ ) = k
r ′−1 r −1
for |r ′| > r;
(A3)
′
for |r | < r.
Averaging over spherical shells on both sides of the relation G(r ′ ) = −∇′ V (r ′ ) yields Gr (r ′ ) = −∇′ Vr (r ′ ). Using Eq. (A3) in this gives Eq. (3).
∗
Electronic address:
[email protected]
1
David J. Griffiths, Introduction to Electrodynamics (Prentice-Hall, New Jersey, 1999) 3rd ed., pp. 173 – 175.
2
John D. Jackson, Classical Electrodynamics (Wiley, New York, 1999) 3rd ed., pp. 148 – 150 and pp. 187 – 188.
3
See e.g., Wolfgang Pauli, Electrodynamics (MIT Press, Cambridge, MA, 1973) pp. 37-39; Paul Lorrain, Dale P. Corson, and Fran¸cois Lorrain, Electromagnetic Fields and Waves (W. H. Freeman, New York, 1988) 3rd ed., pp. 56-57; B. K. P. Scaife, Principles of Dielectrics (Oxford University Press, Oxford, 1989), appendix F; Evaristo Riande and Ricardo Daz-Calleja, Elec-
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trical Properties of Polymers (Marcel Dekker, New York, 2004) p. 44; Saunak Palit, Principles of Electricity and Magnetism (Alpha Science International, Harrow, U.K., 2005) pp. 61-63; Tai L. Chow, Introduction to Electromagnetic Theory: A Modern Perspective (Jones and Bartlett, Boston, 2006) p. 86, problem 9. 4
See, e.g., Ref. 1, pp. 156 – 157, p. 253 and David J. Griffiths, Instructor’s Solutions Manual: Introduction to Electrodynamics (PrenticeHall, Englewood Cliffs, NJ, 1999), pp. 108 – 109; A. Z. Capri and P. V. Panat, Introduction to Electrodynamics (Narosa, New Delhi, 2002), pp. 155158 and pp. 242-246; Ben Yu-Kuang Hu, “Averages of static electric and magnetic fields over a spherical region: A derivation based on the mean-value theorem,” Am. J. Phys. 68, 1058–1060 (2000).
5
Ref. 1, p. 253, problem 5.57.
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The Heavyside function is defined to be Θ(x) = 1 for x > 0, Θ(x) = 0 for x < 0 and Θ(0) = 21 . R Note the difference between the average of a vector field G over a surface A, A dA G/|A| (where R ˆ · G (where n ˆ is a |A| is the area of surface A), and the flux of G through the surface, A dA n
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unit vector perpendicular to area element dA). This paper deals with the former, while Gauss’ law pertains to the latter.
8
See e.g., Ref. 1, p. 254; Ref. 2, p. 186.
9
See e.g., Ref. 1, pp. 114 – 115.
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Figures
FIG. 1: Definition of Gr (r ′ ): the average of the electric field of a unit charge about a spherical surface of radius r, centered at r ′ relative to the charge.
FIG. 2: Integration region for the limits RR R r′ can also be written as 0 dr ′ 0 dr.
RR 0
dr
RR
8
r
dr ′ (grey region). Integration over this region
FIG. 3: Spherical surface over which Gz is averaged to obtain Gr ,z . The hatched strip indicates the area element dA′′ on the sphere with constant Gz (θ) = krz′′ /|r ′′ |3 where rz′′ = r ′ + r cos θ and √ |r ′′ | = r ′2 + r 2 + 2rr ′ cos θ.
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