Abstract

A general formula for calculating radiative fluxes from point sources of radiation incident on spherical objects was derived using some fundamental laws of classical radiometry. This formula was derived in the Cartesian coordinate system, 0xyz, where the coordinates, x, y, and z, determine the position of the spherical object with respect to the point source. The obtained solution was dependent on the radius of the object, and on the function describing the intensity of the radiation. A specific solution for calculating fluxes of isotropic radiations was presented and selected calculations were illustrated graphically.

©2004 Optical Society of America

1. Introduction

Many natural objects and industrial products of spherical shape are irradiated in nature by non-ionizing and ionizing radiation from a single point source or from many point sources. For example, non-ionizing and ionizing radiation from distant stars continuously irradiates planets, planetoids and moons, while a such radiation from non-natural point sources may be used for irradiation of fruits, vegetables and seeds, various microorganisms as well as molecules and atoms when they may be considered as spherical bodies [13]. In addition, a variety of spherical products may be irradiated by artificial point sources of radiation [47].

Treatment of some foods by non-ionizing and ionizing radiation have shown very promising results in destroying harmful pathogens and extending shelf life of some food products without any detectable physical and chemical changes. Such methods of irradiation have been successfully tested in laboratories and should find wide application in the future [1, 37]. Therefore, to study the effect of this radiation on irradiated objects quantitatively accurate analytical methods for estimating the radiative fluxes are necessary.

The main goal of this research was to derive an analytical formula for calculating radiative fluxes emitted by various point source models into a space subtended by a spherical object. As an example, selected results for an isotropic point source in various configurations with respect to the spherical object were calculated and illustrated graphically by three-dimensional surface plots.

2. Basic equations

An infinitesimal element of the radiative flux, dΦP(x, y, z), emitted by a point source, P, located at the point P(0, 0, 0), in the direction determined by the Cartesian coordinates x, y and z is defined by the expression [8]

dΦP(x,y,z)=I(x,y,z)dωP(x,y,z),

where I(x, y, z) is the intensity of the radiation (radiant intensity) emitted by the source P and dωP(x, y, z) is the infinitesimal element of the solid angle subtended by the surface element dS(x, y, z) at the point P(0, 0, 0). The infinitesimal solid angle is defined as [8]

dωP(x,y,z)=dS(x,y,z)cosθr2=zdxdy(x2+y2+z2)32,

where r 2=x 2+y 2+z 2, and θ is the angle between r and the normal to the surface element dS(x, y, z). Substituting Eq. (2) into (1) we get

dΦP(x,y,z)=I(x,y,z)zdxdy(x2+y2+z2)32.

For the center 01 of the spherical object, at the distance z from the origin 0 of the coordinate system 0xyz, we can introduce an additional coordinate system 01 x 1 y 1 z as shown in Fig. 1. In these coordinates, Eq. (3) becomes

 figure: Fig. 1.

Fig. 1. An intersection of a spherical object with the center 01 at a distance r from the origin 0 of the main coordinate system, 0xyz, erected at point source P. The additional coordinate system, 01 x 1 y 1, is erected at the center 01 of the spherical object. The aperture, A, at the distance, d, from the source P limits the width, ρ, of the radiation cross-section in the plane z=constant at zd.

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dΦP(x,y,z;x1,y1)=I(x,y,z;x1,y1)zdx1dy1[(xx1)2+(yy1)2+z2]32.

It is easy to see, that after integration of Eq. (4) with respect to the variables x 1 and y 1 it is possible to obtain expressions for calculating radiative flux as function of the x, y, and z coordinates determining the center 01 of the spherical object.

Transforming coordinates x 1 and x 2 as

x1=x11cosφy11sinφ,
y1=x11sinφy11cosφ,

we can obtain an equation of the ellipse for an external contour of a shadow made by the spherical object on the 01 x 11 y 11 plane (see Fig. 2) given by [9]

(x11c2a1)2a2+y112b2=1,

where a and b denote half-lengths of the major and minor ellipse axes, respectively, and c 2=a 2-b 2.

 figure: Fig. 2.

Fig. 2. Geometrical illustration of the transformation described by Eqs. (5)-(6) and definition of parameters used in Eq. (7) to determine the ellipse made by the external contour of the shadow of the spherical object with the center 01 on the 0′xy plane.

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The geometrical relationships in Fig. 2 lead us to the expression for the half-length of the minor ellipse axis

b=Ra,

and from dependencies between some variables shown in Fig. 3 we can obtain the half-length of the major ellipse axis

a=(a1+a2)/2=Rzr2R2,

where

a1=r2Rzr2R2+Rr2z2,
a2=r2Rzr2R2Rr2z2,

and R is the radius of the spherical object.

 figure: Fig. 3.

Fig. 3. Definition of geometrical parameters used for obtaining Eqs. (9)(11).

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The Jacobian, J, of the transformation described by Eqs. (5)(6) is given by

J=cosφsinφsinφcosφ=1,

and does not vanish for any angle φ. Therefore, Eq. (4) can be rewritten as

dΦP(x,y,z;x11,y11)=I(x,y,z;x11,y11)zdx11dy11(x1122rxyx11+y112+r2)32,

where rxy=(x 2+y 2)1/2.

It should be noted here that Eq. (13) determines the infinitesimal radiative flux incident on a spherical object when the center 01 of this object lies at the distance r>R from the point source P.

3. The total flux of radiation incident from a point source on a spherical object

The total flux, ΦP(x,y,z), of radiation (the total radiative flux) incident from a point source on a spherical object will be given as a double definite integral of dΦP(x,y,z;x 11,y 11) with respect to x 11 and y 11 with the limits for the variable y 11 given as the function of x 11 derived from Eq. (7). The limits for the variable x 11 are determined by the condition ρ-rxy-2a+b 2/a=0 and some geometrical dependencies between the parameters of the ellipse drawn in Fig. 2. Thus, the formula for ΦP(x,y,z) obtains the form

ΦP(x,y,z)={b2a(2a2b2)adx11y11y11I(x,y,z;x11,y11)zdy11(y112+x1122rxy+r2)32,ifrxyρ2a+b2a,b2aρrxydx11y11y11I(x,y,z;x11,y11)zdy11(y112+x1122rxyx11+r2)32,ifrxy>ρ2a+b2a,0,ifrxy>ρ+b2a,

where

y11=ba2[x11(a2b2)/a]2/a,

and

y11=ba2[x11(a2b2)/a]2/a,

It can easily be seen that integrals in Eq. (14) may be further calculated if the radiant intensity function, I(x, y, z, x 11, y 11), is known. For some intensity functions, the inner integrals in Eq. (14) may be evaluated analytically. Then the obtained formula may be simply programmed and used directly in practice.

4. Radiative flux from an isotropic point source incident on a spherical object

In this section, we will present a particular solution to Eq. (14) obtained for radiation emitted by an isotropic point source. The intensity of isotropic radiation from a point source fulfils the relation

4πr2I(x,y,z)=4πz2I(x=0,y=0,z)=4πz2Iz,

where I(x=0, y=0, z)=Iz=constant, and denotes the intensity of radiation emitted in the direction of the z-axis.

Applying Eq. (15) to the space subtended by the spherical object, we get

I(x,y,z;x11,y11)=Izz2y112+x1122rxyx11+r2

and Eq. (14) becomes

ΦP(x,y,z)={Izz3b2a(2a2b2)adx11y11y11dy11(y112+x1122rxyx11+r2)52,ifrxyρ2a+b2a,Izz3b2aρrxydx11y11y11dy11(y112+x1122rxyx11+r2)52,ifrxy>ρ2a+b2a,0,ifrxy>ρ+b2a.

Equation (17) can be integrated analytically with respect to y 11 yielding

ΦP(x,y,z)={Izz3b2a(2a2b2)af(x,y,z;x11)dx11,ifrxyρ2a+b2a,Izz3b2aρrxyf(x,y,z;x11)dx11,ifrxyρ2a+b2a,0,ifrxy>ρ+b2a,

where

f(x,y,z:x11)=2b3b2(2a2b2)+2a(a2b2)x11a2x112[(rxyx11)2+z2]2×4a2b42b6+3a4r2+a[4b2(a2b2)6a3rxy]x11+a2(3a22b2)x112{b4(2a2b2)+r2a4+2a[b2(a2b2)a3rxy]x11+a2(a2b2)x112}3/2.

Evaluating the integrals in Eq. (18) with respect to x 11, we do not obtain any closed form expressions. However, an analytical solution to Eq. (18) exists and may be given by superposition of some elementary functions with elliptic integrals of all three kinds. This solution due to its very long and complicated form is not presented here. For our purposes, Eq. (18) was programmed for numerical simulation of some sample results.

5. Selected results of computer simulation

Selected results of numerical simulation from Eq. (18) obtained using Mathematica 2.2.3 software [10] are presented in Fig. 4(a) and (b). The data illustrate the total radiative flux, ΦP(x,y,z), normalized with respect to the total flux ΦP(x=0,y=0,z) incident on the object at the point 0′(0,0,z), as a dependency on x and y at given z, R, and ρ, Fig. 4(a), and as the function of y and z at given x, R, and ρ, Fig. 4(b). Figure 4(a) shows that the flux ΦP(x,y,z)/ΦP(x=0,y=0,z) depends on x and on y, analogously. However, Fig. 4(b) illustrates that the flux ΦP(x,y,z)/ΦP(x=0,y=0,z) is more sharply dependent on the variation of z than on the variation of y.

 figure: Fig. 4.

Fig. 4. The relative value of the total radiative flux, ΦP(x,y,z)/ΦP(x=0,y=0,z), as a dependency on x and y at z=100, R=1, and ρ=100 (a), and as a dependency on y and z at x=0, R=1, and ρ=100 (b). All geometrical variables (x, y, z, R, and ρ) are expressed in relative units.

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6. Conclusions

We have shown that the total radiative flux, ΦP(x,y,z), incident on spherical objects from any arbitrary point source models may be evaluated from Eq. (14) determined by two double integral expressions dependent on the radiant intensity function I(x, y, z, x 11, y 11). However, no general analytical solution to Eq. (14) can be given until the function I(x, y, z, x 11, y 11) is known. Usually, even then the radiant intensity function is known it is still difficult to obtain an analytical equation in closed form for ΦP(x,y,z) calculation. Therefore, in most cases, numerical procedures for double integral evaluation must be used to estimate the total flux ΦP(x,y,z) of radiation.

For some radiant intensity functions I(x, y, z, x 11, y 11) the inner integrals in Eq. (14) may be expressed in closed form. In these situations it is easy to calculate the flux ΦP(x,y,z) by applying one of many simple numerical procedures for single integral estimation. Sometimes the outer integrals in Eq. (14) may also be expressed simply. In most cases, however, the expressions obtained may be very complicated, so that the numerical procedures of single integral evaluation may be more practical than extended closed analytical solutions.

Equation (14) was then applied to evaluate the radiative flux emitted into the space subtended by the spherical object from an isotropic point source model. For this source model, the solution to Eq. (14) is represented by Eq. (18) described by two single integral expressions. These single integral expressions are calculated as a complex superposition of elementary functions with elliptic integrals of all three kinds. However, this complex solution was not presented in the paper and the flux ΦP(x,y,z) was calculated numerically.

Selected results of numerical simulations were presented in Figs. 4(a) and (b). The three-dimensional relationships of the radiative flux on the spherical object position with respect to the x, y, and z coordinates indicate that the flux is more sensitive to the variation of z than to x or y. These results indicate that the radiative flux changes considerably when the spherical objects moves in the radiation field from a point source. Such variation must be taken into account when systems for spherical object irradiation are designed. The calculation of the radiative flux may also be helpful in estimating the dose of radiation in the processes of spherical food product irradiation if these products are moving in the radiation field from a single point source or from several point sources.

References and links

1. G. Brescia, R. Moreira, L. Braby, and E. Castell-Perez, “Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,” J. Food Engin. 60, 31–39 (2003). [CrossRef]  

2. C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, “Albedo measurements on single particles,” J. Quant. Spectrosc. Radiat. Transfer 55, 673–681 (1996). [CrossRef]  

3. R. Sommer, A. Cabaj, T. Sandu, and M. Lhotsky, “Measurments of UV radiation using suspension of microorganisms,” J. Photochem. Photobiol. B 53, 1–6 (1999). [CrossRef]  

4. J.F. Diehl, “Food irradiation-past, present and future,” Radiat. Phys. Chem. 63, 211–215 (2002). [CrossRef]  

5. R. W. Durante, “Food processors requirements met by radiation processing,” Radiat. Phys. Chem. 63, 289–294 (2002). [CrossRef]  

6. G. W. Gould, “Potential of irradiation as a component of mild combination preservation procedures,” Radiat. Phys. Chem. 48, 366 (1996). [CrossRef]  

7. C. M. Bruhn, “Consumer acceptance of irradiated food: theory and reality,” Radiat. Phys. Chem. 52, 129–133 (1998). [CrossRef]  

8. M. Strojnik and G Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, New York, 2001), pp. 649–699.

9. J. Tallarida, Pocked Book of Integrals and Mathematical Formulas, (CRC, Boca Raton, 1999), Chpt. 4. [CrossRef]  

10. S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass.1993), pp. 44–186.

References

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  1. G. Brescia, R. Moreira, L. Braby, and E. Castell-Perez, “Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,” J. Food Engin. 60, 31–39 (2003).
    [Crossref]
  2. C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, “Albedo measurements on single particles,” J. Quant. Spectrosc. Radiat. Transfer 55, 673–681 (1996).
    [Crossref]
  3. R. Sommer, A. Cabaj, T. Sandu, and M. Lhotsky, “Measurments of UV radiation using suspension of microorganisms,” J. Photochem. Photobiol. B 53, 1–6 (1999).
    [Crossref]
  4. J.F. Diehl, “Food irradiation-past, present and future,” Radiat. Phys. Chem. 63, 211–215 (2002).
    [Crossref]
  5. R. W. Durante, “Food processors requirements met by radiation processing,” Radiat. Phys. Chem. 63, 289–294 (2002).
    [Crossref]
  6. G. W. Gould, “Potential of irradiation as a component of mild combination preservation procedures,” Radiat. Phys. Chem. 48, 366 (1996).
    [Crossref]
  7. C. M. Bruhn, “Consumer acceptance of irradiated food: theory and reality,” Radiat. Phys. Chem. 52, 129–133 (1998).
    [Crossref]
  8. M. Strojnik and G Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, New York, 2001), pp. 649–699.
  9. J. Tallarida, Pocked Book of Integrals and Mathematical Formulas, (CRC, Boca Raton, 1999), Chpt. 4.
    [Crossref]
  10. S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass.1993), pp. 44–186.

2003 (1)

G. Brescia, R. Moreira, L. Braby, and E. Castell-Perez, “Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,” J. Food Engin. 60, 31–39 (2003).
[Crossref]

2002 (2)

J.F. Diehl, “Food irradiation-past, present and future,” Radiat. Phys. Chem. 63, 211–215 (2002).
[Crossref]

R. W. Durante, “Food processors requirements met by radiation processing,” Radiat. Phys. Chem. 63, 289–294 (2002).
[Crossref]

1999 (1)

R. Sommer, A. Cabaj, T. Sandu, and M. Lhotsky, “Measurments of UV radiation using suspension of microorganisms,” J. Photochem. Photobiol. B 53, 1–6 (1999).
[Crossref]

1998 (1)

C. M. Bruhn, “Consumer acceptance of irradiated food: theory and reality,” Radiat. Phys. Chem. 52, 129–133 (1998).
[Crossref]

1996 (2)

G. W. Gould, “Potential of irradiation as a component of mild combination preservation procedures,” Radiat. Phys. Chem. 48, 366 (1996).
[Crossref]

C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, “Albedo measurements on single particles,” J. Quant. Spectrosc. Radiat. Transfer 55, 673–681 (1996).
[Crossref]

Braby, L.

G. Brescia, R. Moreira, L. Braby, and E. Castell-Perez, “Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,” J. Food Engin. 60, 31–39 (2003).
[Crossref]

Brescia, G.

G. Brescia, R. Moreira, L. Braby, and E. Castell-Perez, “Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,” J. Food Engin. 60, 31–39 (2003).
[Crossref]

Bruhn, C. M.

C. M. Bruhn, “Consumer acceptance of irradiated food: theory and reality,” Radiat. Phys. Chem. 52, 129–133 (1998).
[Crossref]

Cabaj, A.

R. Sommer, A. Cabaj, T. Sandu, and M. Lhotsky, “Measurments of UV radiation using suspension of microorganisms,” J. Photochem. Photobiol. B 53, 1–6 (1999).
[Crossref]

Castell-Perez, E.

G. Brescia, R. Moreira, L. Braby, and E. Castell-Perez, “Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,” J. Food Engin. 60, 31–39 (2003).
[Crossref]

Diehl, J.F.

J.F. Diehl, “Food irradiation-past, present and future,” Radiat. Phys. Chem. 63, 211–215 (2002).
[Crossref]

Dröse, G.

C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, “Albedo measurements on single particles,” J. Quant. Spectrosc. Radiat. Transfer 55, 673–681 (1996).
[Crossref]

Durante, R. W.

R. W. Durante, “Food processors requirements met by radiation processing,” Radiat. Phys. Chem. 63, 289–294 (2002).
[Crossref]

Gould, G. W.

G. W. Gould, “Potential of irradiation as a component of mild combination preservation procedures,” Radiat. Phys. Chem. 48, 366 (1996).
[Crossref]

Lhotsky, M.

R. Sommer, A. Cabaj, T. Sandu, and M. Lhotsky, “Measurments of UV radiation using suspension of microorganisms,” J. Photochem. Photobiol. B 53, 1–6 (1999).
[Crossref]

Moreira, R.

G. Brescia, R. Moreira, L. Braby, and E. Castell-Perez, “Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,” J. Food Engin. 60, 31–39 (2003).
[Crossref]

Muinonen, K.

C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, “Albedo measurements on single particles,” J. Quant. Spectrosc. Radiat. Transfer 55, 673–681 (1996).
[Crossref]

Paez, G

M. Strojnik and G Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, New York, 2001), pp. 649–699.

Piironen, J.

C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, “Albedo measurements on single particles,” J. Quant. Spectrosc. Radiat. Transfer 55, 673–681 (1996).
[Crossref]

Sandu, T.

R. Sommer, A. Cabaj, T. Sandu, and M. Lhotsky, “Measurments of UV radiation using suspension of microorganisms,” J. Photochem. Photobiol. B 53, 1–6 (1999).
[Crossref]

Sasse, C.

C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, “Albedo measurements on single particles,” J. Quant. Spectrosc. Radiat. Transfer 55, 673–681 (1996).
[Crossref]

Sommer, R.

R. Sommer, A. Cabaj, T. Sandu, and M. Lhotsky, “Measurments of UV radiation using suspension of microorganisms,” J. Photochem. Photobiol. B 53, 1–6 (1999).
[Crossref]

Strojnik, M.

M. Strojnik and G Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, New York, 2001), pp. 649–699.

Tallarida, J.

J. Tallarida, Pocked Book of Integrals and Mathematical Formulas, (CRC, Boca Raton, 1999), Chpt. 4.
[Crossref]

Wolfram, S.

S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass.1993), pp. 44–186.

J. Food Engin. (1)

G. Brescia, R. Moreira, L. Braby, and E. Castell-Perez, “Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,” J. Food Engin. 60, 31–39 (2003).
[Crossref]

J. Photochem. Photobiol. B (1)

R. Sommer, A. Cabaj, T. Sandu, and M. Lhotsky, “Measurments of UV radiation using suspension of microorganisms,” J. Photochem. Photobiol. B 53, 1–6 (1999).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (1)

C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, “Albedo measurements on single particles,” J. Quant. Spectrosc. Radiat. Transfer 55, 673–681 (1996).
[Crossref]

Radiat. Phys. Chem. (4)

J.F. Diehl, “Food irradiation-past, present and future,” Radiat. Phys. Chem. 63, 211–215 (2002).
[Crossref]

R. W. Durante, “Food processors requirements met by radiation processing,” Radiat. Phys. Chem. 63, 289–294 (2002).
[Crossref]

G. W. Gould, “Potential of irradiation as a component of mild combination preservation procedures,” Radiat. Phys. Chem. 48, 366 (1996).
[Crossref]

C. M. Bruhn, “Consumer acceptance of irradiated food: theory and reality,” Radiat. Phys. Chem. 52, 129–133 (1998).
[Crossref]

Other (3)

M. Strojnik and G Paez, “Radiometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, New York, 2001), pp. 649–699.

J. Tallarida, Pocked Book of Integrals and Mathematical Formulas, (CRC, Boca Raton, 1999), Chpt. 4.
[Crossref]

S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass.1993), pp. 44–186.

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Figures (4)

Fig. 1.
Fig. 1. An intersection of a spherical object with the center 01 at a distance r from the origin 0 of the main coordinate system, 0xyz, erected at point source P. The additional coordinate system, 01 x 1 y 1, is erected at the center 01 of the spherical object. The aperture, A, at the distance, d, from the source P limits the width, ρ, of the radiation cross-section in the plane z=constant at zd.
Fig. 2.
Fig. 2. Geometrical illustration of the transformation described by Eqs. (5)-(6) and definition of parameters used in Eq. (7) to determine the ellipse made by the external contour of the shadow of the spherical object with the center 01 on the 0′xy plane.
Fig. 3.
Fig. 3. Definition of geometrical parameters used for obtaining Eqs. (9)(11).
Fig. 4.
Fig. 4. The relative value of the total radiative flux, ΦP (x,y,z)/ΦP(x=0,y=0,z), as a dependency on x and y at z=100, R=1, and ρ=100 (a), and as a dependency on y and z at x=0, R=1, and ρ=100 (b). All geometrical variables (x, y, z, R, and ρ) are expressed in relative units.

Equations (21)

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d Φ P ( x , y , z ) = I ( x , y , z ) d ω P ( x , y , z ) ,
d ω P ( x , y , z ) = d S ( x , y , z ) cos θ r 2 = z d x d y ( x 2 + y 2 + z 2 ) 3 2 ,
d Φ P ( x , y , z ) = I ( x , y , z ) z d x d y ( x 2 + y 2 + z 2 ) 3 2 .
d Φ P ( x , y , z ; x 1 , y 1 ) = I ( x , y , z ; x 1 , y 1 ) z d x 1 d y 1 [ ( x x 1 ) 2 + ( y y 1 ) 2 + z 2 ] 3 2 .
x 1 = x 11 cos φ y 11 sin φ ,
y 1 = x 11 sin φ y 11 cos φ ,
( x 11 c 2 a 1 ) 2 a 2 + y 11 2 b 2 = 1 ,
b = R a ,
a = ( a 1 + a 2 ) / 2 = R z r 2 R 2 ,
a 1 = r 2 R z r 2 R 2 + R r 2 z 2 ,
a 2 = r 2 R z r 2 R 2 R r 2 z 2 ,
J = cos φ sin φ sin φ cos φ = 1 ,
d Φ P ( x , y , z ; x 11 , y 11 ) = I ( x , y , z ; x 11 , y 11 ) z d x 11 d y 11 ( x 11 2 2 r xy x 11 + y 11 2 + r 2 ) 3 2 ,
Φ P ( x , y , z ) = { b 2 a ( 2 a 2 b 2 ) a d x 11 y 11 y 11 I ( x , y , z ; x 11 , y 11 ) z d y 11 ( y 11 2 + x 11 2 2 r xy + r 2 ) 3 2 , if r xy ρ 2 a + b 2 a , b 2 a ρ r xy d x 11 y 11 y 11 I ( x , y , z ; x 11 , y 11 ) z d y 11 ( y 11 2 + x 11 2 2 r xy x 11 + r 2 ) 3 2 , if r xy > ρ 2 a + b 2 a , 0 , if r xy > ρ + b 2 a ,
y 11 = b a 2 [ x 11 ( a 2 b 2 ) / a ] 2 / a ,
y 11 = b a 2 [ x 11 ( a 2 b 2 ) / a ] 2 / a ,
4 π r 2 I ( x , y , z ) = 4 π z 2 I ( x = 0 , y = 0 , z ) = 4 π z 2 I z ,
I ( x , y , z ; x 11 , y 11 ) = I z z 2 y 11 2 + x 11 2 2 r xy x 11 + r 2
Φ P ( x , y , z ) = { I z z 3 b 2 a ( 2 a 2 b 2 ) a d x 11 y 11 y 11 d y 11 ( y 11 2 + x 11 2 2 r xy x 11 + r 2 ) 5 2 , if r xy ρ 2 a + b 2 a , I z z 3 b 2 a ρ r xy d x 11 y 11 y 11 d y 11 ( y 11 2 + x 11 2 2 r xy x 11 + r 2 ) 5 2 , if r xy > ρ 2 a + b 2 a , 0 , if r xy > ρ + b 2 a .
Φ P ( x , y , z ) = { I z z 3 b 2 a ( 2 a 2 b 2 ) a f ( x , y , z ; x 11 ) d x 11 , if r xy ρ 2 a + b 2 a , I z z 3 b 2 a ρ r xy f ( x , y , z ; x 11 ) d x 11 , if r xy ρ 2 a + b 2 a , 0 , if r xy > ρ + b 2 a ,
f ( x , y , z : x 11 ) = 2 b 3 b 2 ( 2 a 2 b 2 ) + 2 a ( a 2 b 2 ) x 11 a 2 x 11 2 [ ( r x y x 11 ) 2 + z 2 ] 2 × 4 a 2 b 4 2 b 6 + 3 a 4 r 2 + a [ 4 b 2 ( a 2 b 2 ) 6 a 3 r x y ] x 11 + a 2 ( 3 a 2 2 b 2 ) x 11 2 { b 4 ( 2 a 2 b 2 ) + r 2 a 4 + 2 a [ b 2 ( a 2 b 2 ) a 3 r x y ] x 11 + a 2 ( a 2 b 2 ) x 11 2 } 3 / 2 .

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