Electrostatics in the presence of dielectrics: The benefits of treating the induced surface charge density directly

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. (C) 2004 American Association of Physics Teachers.