A new method is presented for solving electrostatic boundary value problems with dielectrics or conductors and is applied to systems with spherical geometry. The strategy of the method is to treat the induced surface charge density as the variable of the boundary value problem. Because the potential is expressed directly in terms of the induced surface charge density, the potential is automatically continuous at the boundary. The remaining boundary condition produces an algebraic equation for the surface charge density, which when solved leads to the potential. The surface charge method requires the enforcement of only one boundary condition, and produces the induced surface charge in addition to the potential with no additional labor. The surface charge method also can be applied in nonspherical geometries and provides a starting place for efficient numerical solutions.

1.
Yi-Kuo
Yu
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On a class of integrals of Legendre polynomials with complicated arguments—with applications in electrostatics and biomolecular modeling
,”
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(
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2.
The following is a selection of textbooks that discuss electrostatics, but do not mention the surface charge method. D. J. Griffiths, Introduction to Electrodynamics (Prentice Hall, Englewood Cliffs, NJ, 1981);
J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975);
P. Lorrain, D. R. Corson, and F. Lorrain, Electromagnetic Fields and Waves, 3rd ed. (W. H. Freeman, New York, 1988);
G. L. Pollack and D. R. Stump, Electromagnetism (Addison Wesley, San Francisco, 2002);
M. Schwartz, Principles of Electrodynamics (Dover, New York, 1987);
O. D. Jefimenko, Electricity and Magnetism, 2nd ed. (Electret Scientific Co., Star City, WV, 1989);
J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941);
W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, MA, 1955);
L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, New York, 1984).
3.
T.
Sometani
and
K.
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(
1977
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Rosalind
Allen
,
Jean-Pierre
Hansen
, and
Simone
Melchionna
, “
Electrostatic potential inside ionic solutions confined by dielectrics: a variational approach
,”
Phys. Chem. Chem. Phys.
3
,
4177
4186
(
2001
).
5.
In a uniform linear dielectric with dielectric constant ε and free and total charge densities ρf and ρ, we have 4πρf=∇⋅D⃗=∇⋅εE⃗=ε∇⋅E⃗=ε4πρ, the first and last members of which imply that ρf=ερ. The first equality is the differential form of Gauss’s law for the displacement field D⃗, the second equality follows from D⃗=εE⃗ for a linear dielectric, the third equality is a consequence of the spatial uniformity of ε, and the fourth equality is a result of the differential form of Gauss’s law for the electric field E⃗. At the boundary between two distinct linear dielectric materials, the third equality does not hold because the dielectric constant is not locally constant there. Therefore, the total charge density may be nonzero at the boundary even if the free charge density vanishes there. In the special case where the free charge distribution is in the form of a point charge qf, the total charge (except that at the boundary) is qf/ε.
6.
The total free charge is Q0. The monopole contribution to the electric displacement outside the void is D⃗mp=(Q0r̂)/r2. Because D⃗mpexE⃗ in the infinite dielectric, E⃗mp=(Q0r̂)/(εexr2) and consequently Φmp=Q0/(εexr). The monopole moment is identified as Q0ex which must be the total charge of the charge distribution.
7.
If we let either dielectric constant go to infinity, we recover the case of a conductor. If the other dielectric constant is set to unity, corresponding to vacuum, we recover a situation commonly discussed in textbooks. See, for example, Griffiths, Ref. 2, pp. 85–93.
8.
Indeed, the dielectric constants of the various finite pieces of material need not all be the same, but in order to avoid excessively elaborate notation at this stage, attention will be restricted to the case indicated. The generalization is straightforward.
9.
See nearly any of the books in Ref. 2. Griffiths contains a notably lucid explanation.
10.
In the degenerate case that the three points are collinear, the z axis can be taken to be perpendicular to the line that includes all three points. In this case, θ12=π/2.
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