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In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
Diagram illustrating the image method for Laplace's equation for a sphere of radius R. The green point is a charge q lying inside the sphere at a distance p from the origin, the red point is the image of that point, having charge −qR/p, lying outside the sphere at a distance of R 2 /p from the origin. The potential produced by the two charges ...
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical times , it has been known that some materials, such as amber , attract lightweight particles after rubbing .
In fact, virtually all the mathematics found in recent papers was already done by Chester Snow. This is found in his book titled Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory, National Bureau of Standards Applied Mathematics Series 19, 1952. Look specifically on pages 228-263.
In potential theory, an area of mathematics, a double layer potential is a solution of Laplace's equation corresponding to the electrostatic or magnetic potential associated to a dipole distribution on a closed surface S in three-dimensions.
In mathematics and mathematical physics, potential theory is the study of harmonic functions.. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which ...
In the case where there is no source term (e.g. vacuum, or paired charges), these potentials obey Laplace's equation: = For a distribution of mass (or charge), the potential can be expanded in a series of spherical harmonics , and the n th term in the series can be viewed as a potential arising from the 2 n -moments (see multipole expansion ).
Earnshaw's theorem states that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges. This was first proven by British mathematician Samuel Earnshaw in 1842. It is usually cited in reference to magnetic fields, but was first applied to ...