Averaging of harmonic physical fields over an annular region enclosing field sources

Abstract

Fields such as temperature, current density, and static electromagnetic fields in regions with no field sources are harmonic functions that satisfy the Laplace equation. Such functions on a sphere have a well-known mean value property. A new mean value property is derived for fields that are harmonic on an annular region, with the field sources enclosed by the inner boundary. Some examples are discussed.

I. INTRODUCTION

The average of static electric and magnetic fields over a spherical region is discussed in electromagnetic (EM) textbooks,^1–3 and recently in a paper by Hu.⁴ In a region with no field sources,⁵ the mean values of the EM fields over a spherical surface or shell are equal to the field values at the center of the sphere, r=0:

$${\langle E(r)\rangle }_{\text{sp}}=E(0),$$ (1a)

$${\langle B(r)\rangle }_{\text{sp}}=B(0),$$ (1b)

where 〈 〉_sp denotes the spherical mean value, that is, the mean value of the fields over the spherical shell or surface. A simple proof of Eq. (1) was given in Ref. 4 by applying the mean value theorem of harmonic functions⁶ to a scalar potential and then taking the spatial gradients. A harmonic function is defined as any solution of the Laplace equation (for any boundary conditions). In fact, Eq. (1) can be obtained directly by applying the mean value theorem to EM fields because these vector fields are harmonic functions throughout the sphere as well.^7,8

We will refer to harmonic functions in space as harmonic fields.⁹ Harmonic fields of physical interest include scalars such as temperature, illumination,¹⁰ and conservative potentials (for example, electric, magnetic, and gravitational potentials); vector fields such as current density, irrotational flow velocity,¹⁰ static EM fields and vector potentials, magnetic field gradient tensors,¹¹ and gravitational fields. These vector fields have zero divergence and curl in the absence of sources, and hence are harmonic fields. [For an arbitrary vector A, we have ∇×(∇×A)=∇(∇·A)−∇²A. If ∇×A=0 and ∇·A=0, we have ∇²A=0, and hence A is a harmonic field.]

The mean value property in Eq. (1) can be generalized to a variety of harmonic fields u (which can be either a scalar or a vector in multidimensional space),

$${\langle u(r)\rangle }_{\text{sp}}=u(0).$$ (2)

Equation (2) indicates that the value of a field at a point can be obtained from its surrounding values on a spherical region centered at this point. This relation is particularly useful when the location of interest is not accessible by direct measurement. Another application of this relation is to improve the precision of a measurement by replacing the center field values by the spherical mean value, which could significantly reduce the random noise.^8,12

When there are field sources inside the sphere, the Laplacian of u(r) is nonzero at the locations of the field sources. Then u will not be harmonic everywhere throughout the sphere, and the validity of Eq. (2) is questionable. However, prior work^1,2,4 has demonstrated that the mean values of static EM fields over an entire sphere are equal to the field values of the dipoles inside the sphere, that is,

$${\langle E(r)\rangle }_{\text{entire sphere }}=-\frac{p}{4\pi {ϵ}_0{R}^3},$$ (3a)

$${\langle B(r)\rangle }_{\text{entire sphere }}=\frac{{\mu }_{0}m}{2\pi R^3},$$ (3b)

where R is the radius of the sphere, ϵ₀ is the electric dielectric constant in a vacuum, μ₀ is the magnetic permeability of vacuum, and p and m are the net electric or magnetic dipole moment within the sphere, respectively.

In this paper we consider the average of harmonic fields over an annular region, for example, a spherical shell in three-dimensional space. The field sources exist within the shell’s inner boundary. A general mean value property will be derived for fields that are harmonic on the annular region. Without loss of generality, a scalar harmonic field will be considered. However, the discussion can be readily extended to any vector field whose components are harmonic scalar functions. Potential applications of this property will be illustrated by a few examples.

II. MEAN VALUE PROPERTIES ON AN ANNULAR REGION

An annular region Ω centered at the origin r=0 in n-dimensional space (n⩾2) is defined as the region enclosed by an inner boundary of constant radius, r₁, and an outer boundary of constant radius, r₂ (r₂⩾r₁) (see Fig. 1). When r₁=r₂=r, Ω becomes a spherical surface, S; S reduces to a circle C when n=2. The region within the inner boundary where the field sources are enclosed is designated as Λ. The field generated by these sources, u(r), is assumed to be harmonic in Ω.

Fig. 1.

Harmonic field u(r) in an annular region Ω (dotted area, O is the center, r₁<r<r₂). The field sources are in the region Λ(r<r₁). S is a surface at radius r.

From harmonic function theory,⁶ u within Ω can be expanded in a Laurent series, that is,

$$u(r)=\sum _{k=0}^{\infty }{p}_k(r)+\sum _{k=0}^{\infty }\frac{q_k(r)}{r^{2k+n-2}}(n>2),$$ (4a)

$$u(r)=\sum _{k=0}^{\infty }{p}_k(r)+q_0\text{ log }r+\sum _{k=1}^{\infty }\frac{q_k(r)}{r^{2k}}(n=2),$$ (4b)

where p_k and q_k are homogeneous harmonic polynomials of degree k. The harmonic polynomials satisfy the Laplace equation in space. They are zero at r=0, except that p₀ and q₀ are constant (see Appendix A). If the field u is averaged over S (n>2) or C (n=2), the mean values of all the k≠0 harmonic polynomials will be zero due to the mean value property of harmonic functions [see Eq. (A3)]. The spherical mean value of u depends only on the contribution of the constant term, and a rⁿ⁻² term for n>2 or a logarithmic term for n=2. That is (see Appendix A),

$${\langle u(r)\rangle }_S=p_0+\frac{q_0}{r^{n-2}}(n>2),$$ (5a)

$${\langle u(r)\rangle }_C=p_0+q_0\text{ log }r(n=2).$$ (5b)

This result can be extended to an average over the whole region Ω by integrating Eq. (5) over the radius r from r₁ to r₂ (see Appendix A). The mean value of the field u on Ω still depends only on p₀, q₀, and the radii r₁ and r₂ [see Eq. A5)]. If p₀ and q₀ are both zero, the mean value of u is

$${\langle u\rangle }_{C,S\text{ or }\Omega }=0.$$ (6)

Equation (6) can be directly applied to static EM fields generated by multipole sources in the region Λ. As pointed out in Sec. I, these fields are harmonic in the region Ω with no sources. Therefore, E and B can be written in the form of Eq. (4). However, multipole static EM fields are known to vary inversely as r⁻ⁿ (r>r₁) with n⩾2. Therefore, the constants p₀ and q₀ in Eq. (4) must be zero. With the help of Eqs. (5) or (6), we have

$${\langle E\rangle }_{S\text{ or }\Omega }=0,$$ (7a)

$${\langle B\rangle }_{S\text{ or }\Omega }=0,$$ (7b)

if all the field sources are contained in the region Λ.

Equation (7) can be confirmed in a different way by examining the EM field distribution in three-dimensional space (see Appendix B). The apparent discrepancy of Eqs. (7) and (3) can be explained by the fact that the fields in Eq. (3) are averaged over the entire sphere whereas those in Eq. (7) are averaged only over an annular region, excluding the field sources. As shown below, Eq. (7) can be derived directly from Eq. (3). If both sides of Eq. (3) are multiplied by the volume of the sphere 4πR³/3, then

$${\langle E(r)\rangle }_{\text{entire sphere }}\frac{4\pi R^3}{3}=-\frac{p}{3{ϵ}_0},$$ (8a)

$${\langle B(r)\rangle }_{\text{entire sphere }}\frac{4\pi R^3}{3}=\frac{2{\mu }_{0}m}{3}.$$ (8b)

The product of the spherical mean value and the sphere volume, shown on the left-side of Eq. (8), is equivalent to the integration of the fields over the entire sphere of radius R. Equation (8) demonstrates that the integration over an entire sphere is independent of the radius. If the two concentric spheres with radii r₁ and r₂ (r₁<r₂) enclose the same field sources, and thus the same dipole moments, then the integration over the sphere of radius r₁ would be equal to the integration over the sphere of radius r₂. The integration over the shell Ω (r₁<r<r₂) is identical to the difference of the integration over these two spheres, which gives a null value and leads to Eq. (7). (The integration over S can be considered as a limiting case of integration over the shell Ω when r₁ becomes close to r₂.)

III. POSSIBLE APPLICATIONS

A. Measurement of p₀ and q₀

The values of p₀ and q₀ could reflect important characteristics of the fields. They can be obtained from Eq. (5) by measuring u(r) and finding the r-dependence of 〈u(r)〉_S. For example, suppose that u(r) is the electrostatic potential generated by a charge distribution in the region Λ. Q is the total net charge, and n=3. From Eq. (5a) we have

$${\langle u(r)\rangle }_S=p_0+\frac{q_0}{r}.$$ (9)

If we have adopt the convention that the potential at infinity is zero, p₀=0, we obtain

$${\langle u(r)\rangle }_S=\frac{q_0}{r}.$$ (10a)

Here q₀ should be equal to Q/4πϵ_0, because u should become increasingly close to the potential field of a single charge Q as r increases. Thus

$${\langle u(r)\rangle }_S=\frac{Q}{4\pi {ϵ}_{0}r}.$$ (10b)

Equation (10b) shows that the spherical averaging procedure smoothes out the contributions of higher-order multipole fields. Regardless of the spatial charge distribution of the field sources, the net charge Q can be obtained rigorously from the mean value of the field on the spherical surface S, which is a distance r away from the sources such that the surface encloses the entire charge distribution. Because S can be near the field sources, obtaining Q from the mean value of the fields over S is more practical than obtaining Q from the conventional measurement of the farfields. Gauss’ theorem,²

$${\iint }_{S}E\cdot \widehat{\mathbf{n}}ds=\frac{Q}{{ϵ}_0},$$ (11)

can also be used to find Q by measuring the vector field E on the surface S ($\widehat{\mathbf{n}}$ is the unit normal vector of the surface). In comparison, the mean value method has the advantage of using only a scalar quantity.

In the same way, this method can be applied to find the temperature distribution generated by heat sources in a homogeneous medium.¹⁰ The total amount of heat loss or generation can be obtained exactly from the spherical mean of the temperature on a spherical surface enclosing the sources, without knowing their spatial distribution and the strength of each source.

B. In situ quantification of external field

Thus far, all the field sources were assumed to be located within the region Λ. In general, some external field sources can exist and generate an external field u_e superimposed on the internal field u_i generated by the internal sources. The determination of u_e is useful for applications such as the study of electromagnetic polarization in a medium. What is usually measured is the total field u_t:

$$u_t(r)=u_e(r)+u_i(r).$$ (12)

If we take the spherical mean of both sides of Eq. (12), we obtain

$${\langle u_t(r)\rangle }_{S\text{ or }\Omega }={\langle u_e(r)\rangle }_{S\text{ or }\Omega }+{\langle u_i(r)\rangle }_{S\text{ or }\Omega }.$$ (13)

Because the sources for u_e are located outside the outer boundary of Ω, it is reasonable to assume that u_e is harmonic in both Ω and Λ. With the use of Eq. (2), we have

$${\langle u_e(r)\rangle }_{S\text{ or }\Omega }=u_e(0).$$ (14)

The field u_i is usually assumed to be zero at infinity (p₀=0), and hence Eq. (10a) applies to u_i. If Eqs. (10a) and (14) are substituted into Eq. (13), we obtain

$${\langle u_t(r)\rangle }_s=u_e(0)+\frac{q_0(0)}{r}.$$ (15)

Here u_e(0) and q₀ can be obtained by fitting 〈u_t(r)〉_s to 1/r. If p₀=q₀=0, we can use Eq. (14) for 〈u_e(r)〉_{S or Ω} and Eq. (6) for 〈u_i(r)〉_{S or Ω}, and write Eq. (13) as

$$u_e(0)={\langle u_t(r)\rangle }_{S\text{ or }\Omega }.$$ (16)

Equations (15) and (16) indicate that the external field at a point inside an object can be obtained rigorously from the surrounding field values on spherical surfaces or shells enclosing the object. The internal field u_i at r=0 can then be obtained from Eq. (12). Thus, the field contribution from external and internal sources, u_e and u_i, respectively, can be separately found at any field location.

The advantages of this method of external field determination can be demonstrated by considering static EM polarization.² The external field polarizes an object proportional to its electric/magnetic susceptibility and also induces the depolarizing field. The field that is measured usually includes the contribution from the external field, the depolarizing field (the internal field), and the local polarization. The goal is to find the susceptibility from the field distribution. In some cases the external field is sufficiently homogeneous so that it can be represented by a constant. In other cases, such as in magnetic resonance imaging (MRI), the external field inhomogeneity can be comparable to the magnetization induced in diamagnetic and paramagnetic substances.¹³ Therefore, to accurately determine the susceptibility, it is important to know the value of the exact external field for a location inside the object.¹⁴ The usual approximation is to measure the field at the same location after taking the object away or replacing it with a reference object, assuming that the object removal or replacement has little effect on the external field distribution. However, this approach is not always practical and the assumption is not valid for strongly coupled systems (for example, ferromagnetic objects) in which significant mutual polarization couplings exist between the object and the external objects or field sources.

Equation (16) directly applies to static EM fields, and the external field can be determined by the spherical mean value of the measured field on a spherical surface or shell in the surrounding medium. This method is useful and convenient because it is an in situ approach. The object does not have to be removed or replaced by a reference object. The method does require much field distribution information at locations on the spherical shell or surface. The field distribution can be mapped with modern imaging techniques such as MRI. With the in situ external field determination, a rigorous methodology has been successfully developed that can quantify the magnetic susceptibility for arbitrarily shaped objects in inhomogeneous fields.¹⁴ In principle, a similar methodology can be applied to the measurement of electric susceptibility.

C. Precision improvement

As shown in previous work,^8,12 we can significantly reduce the measurement noise by applying the mean value method to physical quantities such as temperature and magnetic fields that satisfy the Laplace equation. The method is to replace field the value at every point by its spherical mean value—the mean value of the field u over a spherical region Ω centered at this point. The noise in the data is reduced by a factor of the square root of the total number of the data points used in averaging. For a 3-D image data, when averaging over an entire sphere of a radius of 100 points, the noise reduction is about 2000 times! For example, the precision of conventional MRI field mapping is about 10⁻⁸ Tesla (T). In comparison, the mapping precision has been increased to 10⁻¹¹–10⁻¹² T using the spherical mean value method over a sphere whose radius is about 20 points.⁸ The difference between the spherical mean value method and conventional simple averaging lies in the fact that the mean value property on a spherical region is an inherent characteristic of harmonic fields. The spherical mean value method is a theoretically rigorous approach rather than an approximation, as is usually the case for simple averaging.

Similar improvements can be achieved using the mean value property on an annular region. The constant q₀ in Eq. (10) and the external field in Eq. (16) can be determined by averaging a number of field points, which would substantially reduce the random noise present in the raw field measurements. It is expected that the quantities determined in this way have a noise level much lower than that of the raw field measurements.

IV. CONCLUSION

A new general mean value property has been derived for harmonic fields in an annular region. Although static electromagnetic fields have been discussed as typical examples, this property could be applied to other physical fields as well. Provided that physical quantities such as temperature, current density, flow, and the gravitational force are harmonic fields, the mean value properties of harmonic fields are expected to have many applications. Suggested problems for students are given in Appendix C.

APPENDIX A: THE MEAN VALUE OF HARMONIC FUNCTIONS ON AN ANNULAR REGION

As described in Eq. (4), harmonic functions can be expanded into a Laurent series consisting of various orders of homogeneous harmonic polynomials. To illustrate the property of these polynomials, we will consider the homogeneous polynomials p_m in three-dimensional space (n=3). In general,

$$p_m(r)=\sum _{i+j+k=m}{c}_{ijk}{x}^i{y}^j{z}^k,$$ (A1)

where x, y, and z are the coordinates of r along three orthogonal directions, c_ijk are constant coefficients, and i, j, k, m are non-negative integers. More specifically,

$$\begin{array}{l}{p}_0=\text{const},\hfill \\ p_1=c_{100}x+c_{010}y+c_{001}z,\hfill \\ p_2=c_{200}{x}^2+c_{020}{y}^2+c_{002}{z}^2+c_{110}xy+c_{011}yz+c_{101}xz,\hfill \\ \cdots .\hfill \end{array}$$ (A2)

The coefficients c_ijk must satisfy certain relations to make p_m harmonic. For example, if ∇²p₂=2c₂₀₀+2c₀₂₀+2c₀₀₂=0, then c₂₀₀+c₀₂₀+c₀₀₂=0. It is obvious that for all m≠0, p_m(0)=0, that is, the homogeneous polynomials except p₀ are zero at the origin (r=0). Based on the mean value property of harmonic functions,⁶

$${\langle p_m(r)\rangle }_{S\text{ or }C}=p_m(0)=\{\begin{array}{ll}0,\hfill & \text{ for }m\ne 0,\hfill \\ p_0,\hfill & \text{ for }m=0.\hfill \end{array}$$ (A3)

This result applies to any harmonic homogeneous polynomial in a n-dimensional space (n⩾2).

To prove Eq. (5), the spherical mean value is taken for each term on both sides of Eq. (4). With the use of Eq. (A3) for n>2, we obtain

$$\begin{gathered} {\langle u(r)\rangle }_S=\sum _{k=0}^{\infty }{\langle p_k(r)\rangle }_S+\sum _{k=0}^{\infty }{\langle \frac{q_k(r)}{r^{2k+n-2}}\rangle }_S \\ =p_0+\sum _{k=0}^{\infty }\frac{{\langle q_k(r)\rangle }_S}{r^{2k+n-2}}=p_0+\frac{q_0}{r^{n-2}}, \end{gathered}$$ (A4a)

and for n=2,

$$\begin{gathered} {\langle u(r)\rangle }_C=\sum _{k=0}^{\infty }{\langle p_k(r)\rangle }_C+{\langle q_0(r)\text{log }r\rangle }_C+\sum _{k=1}^{\infty }{\langle \frac{q_k(r)}{r^{2k}}\rangle }_C \\ =p_0+\text{log }r{\langle q_0(r)\rangle }_C+\sum _{k=1}^{\infty }\frac{{\langle q_k(r)\rangle }_C}{r^{2k}} \\ =p_0+q_0\text{ log }r, \end{gathered}$$ (A4b)

where the denominators can be taken out of the brackets because they are constant on the surface S or the circle C. The mean value of the field u over S or C depends only on p₀, q₀, and the radius r. This result can be extended to an annular region Ω by the integration of the right-sides of Eq. (A4) over the radius r over the interval [r₁,r₂]. For n>2,

$$\begin{gathered} {\langle u(r)\rangle }_V=\frac{1}{V}{\int }_{r_1}^{r_2}\left(p_0+\frac{q_0}{r^{n-2}}\right)\beta r^{n-1}dr \\ =p_0+\frac{q_0\beta }{V}{\int }_{r_1}^{r_2}rdr=p_0+\frac{q_0\beta }{2V}\left(r_2^2-r_1^2\right), \end{gathered}$$ (A5a)

where V is the volume of Ω, $V={\int }_{r_1}^{r_2}\beta r^{n-1}dr$; β is a constant dependent on n (β=4 π for n=3). For n=2,

$$\begin{gathered} {\langle u(r)\rangle }_{\Omega }=\frac{1}{A}{\int }_{r_1}^{r_2}\left[p_0+q_0\text{ log }(r)\right]2\pi rdr \\ =p_0+\frac{2q_0}{\left(r_2^2-r_1^2\right)}{\int }_{r_1}^{r_2}r\text{ log }(r)dr \\ =p_0+q_0\left(\text{log }{r}_2+\frac{\text{log }{r}_2-\text{log }{r}_1}{\left(r_2^2-r_1^2\right)}-\frac{1}{2}\right), \end{gathered}$$ (A5b)

where A is the area of Ω. In conclusion, the mean value of the field u over the annular region depends only on p₀, q₀, and the radii r₁ and r₂.

APPENDIX B: SPHERICAL MEAN VALUE OF ELECTROSTATIC AND MAGNETOSTATIC FIELDS ON AN ANNULAR REGION

The region Λ is enclosed by the inner boundary (r=r₁) of an annular region Ω(r₂>r>r₁). The z component of the electrostatic field generated by an electric charge distribution in Λ, E_z, satisfies the Laplace equation in the region Ω. This is a direct result of the EM wave equations for static fields in a space with no field sources. Laplace’s equation can be written in spherical coordinates as

$$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial E_z}{\partial r}\right)+\frac{1}{r^2\text{sin}\theta }\frac{\partial }{\partial \theta }\left(\text{sin}\theta \frac{\partial E_z}{\partial \theta }\right)+\frac{1}{r^2\text{sin}\theta }\frac{{\partial }^2{E}_z}{\partial {\varphi }^2}=0\left(r>r_1\right).$$ (B1)

If we use the conventional treatment of electrostatic scalar potentials,² the solution of Eq. (B1) takes the form

$$E_z(r,\theta )=\sum _{l=0}^{\infty }\sum _{m=-l}^l\left(A_{lm}{r}^l+\frac{B_{lm}}{r^{l+1}}\right)Y_l^m(\text{cos}\theta ).$$ (B2)

Here A_lm and B_lm are constants, and $Y_l^m$ are the spherical harmonics satisfying

$$\iint Y_l^m(\text{cos}\theta )\text{sin}\theta d\theta d\varphi =0,\text{ if }(l,m)\ne (0,0).$$ (B3)

It is well known that when r>r_1, the multipole fields generated by the charge distribution in Λ must vary as 1/r² or higher inverse orders of the distance. Because A_lm are the coefficients for terms r^l (l>0), and B₀₀ is the coefficient for the r⁻¹ term, all the A_lm and B₀₀ must be zero. The spherical mean value of E_z over the surface S is

$$\begin{gathered} {\langle E_z\rangle }_S=\frac{1}{4\pi r^2}\int E_z(r,\theta )ds \\ =\frac{1}{4\pi r^2}\sum _{l=1}^{\infty }\left(\frac{B_{lm}}{r^{l+1}}\right)\cdot \iint Y_l^m(\text{cos}\theta )r^2\text{sin}\theta \cdot d\theta d\varphi \\ =\frac{1}{4\pi }\sum _{l=1}^{\infty }\left[\left(\frac{B_{lm}}{r^{l+1}}\right)\cdot \iint Y_l^m(\text{cos}\theta )\text{sin}\theta d\theta d\varphi \right] \\ =0, \end{gathered}$$ (B4)

where Eq. (B3) was used in the last step. Similar identities hold for the x and y components of the electric fields, and thus

$${\langle E\rangle }_S=0.$$ (B5)

Alternatively Eq. (B5) can be obtained without using the general solution of Eq. (B1). If one applies the spherical surface average to both sides of Eq. (B1), the left second and third terms are obviously zero. Thus we obtain

$$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial {\langle E_z\rangle }_S}{\partial r}\right)=0,$$ (B6)

and the solution is

$${\langle E_z\rangle }_S=p_0+\frac{q_0}{r}.$$ (B7)

We can see that Eq. (B7) indicates an alternative way of deriving Eq. (5) without using the Laurent series expansion, Eq. (4). Obviously p₀ and q₀ in Eq. (B7) are zero because the electric field E_z generated by the field sources within Λ behaves only as 1/r² or higher order when r>r₁. In this way Eqs. (B4) and (B5) can be derived. Similarly, the magnetic field B generated by multipole field sources in Λ also behaves as 1/r² or higher order, and thus

$${\langle B\rangle }_S=0.$$ (B8)

APPENDIX C: SUGGESTED PROBLEMS

We can test the ideas in this paper by using a computer to generate a localized electric charge distribution in three-dimensional space, and to calculate the electric field and scalar potential on spherical surfaces. The field sources can be single charges, dipoles, higher order multipoles, or any combination of them. Below are some suggested problems. Similar problems can be designed in two-dimensional space.

1. Show that the mean value of the electric field on a spherical surface S enclosing all the sources is zero [Eq. (7a)]. Vary the location of these sources within the sphere, and see if the mean value on S stays same.

2. Validate Eq. (10b) by plotting the mean value of the scalar potential field over S against the radius of S. Use either Eq. (10b) or Eq. (11) to determine the total net charge, and compare the result with the value preassigned by the computer.

3. Add random noise to the field data and see how it may deviate the results in Problems (1) and (2). Show that the spherical mean value can significantly reduce the deviations as the radius of S becomes larger.

4. Choose an annular region enclosing the field sources, that is, a spherical shell Ω with inner radius r₁ and outer radius r₂. Validate a special case of Eq. (A5a) (p₀=0, q₀=Q/4πϵ₀) by calculating the mean value of the scalar potential over Ω with r₁ or r₂ varying.