In the simulation of few-body problems in Chapter 4, gravity was the sole interaction at large distances. If the particles are charged, they will interact via electromagnetic forces which are much stronger than gravity at the same distance. For example, the ratio of the Coulomb force to gravity between a proton and an electron is about 10^{39} at a given separation, making gravity utterly negligible. In fact, macroscopic properties of all ordinary matter are determined by electromagnetic interactions. Forces such as tension in a string (Section 6.3), spring forces, friction, etc., are all manifestation of electromagnetic forces between atoms.

Like gravity, electromagnetic interactions occur through action at a distance via electromagnetic fields described by classical field equations, the Maxwell’s equations. In this chapter, we are interested in the simulation of electric potentials and fields in electrostatics and propagation of electromagnetic waves. Both scalar and vector fields in electromagnetism will be studied by solving the appropriate PDEs first encountered in Chapter 6. We discuss two mesh-based methods for solving the 2D Poisson and Laplace equations and related boundary value problems: an iterative self-consistent (relaxation) method, and a noniterative finite element method first introduced for 1D problems in Chapter 6. Complementing mesh-based methods, we introduce a meshfree method using radial basis functions for solving PDEs. Finally, ...

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