Significance of laplace and poisson equation
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The Laplace/Poisson Equasions are the Helmholts equations when the time derivative is zero (f=0) . For this case there is no dependance between the magnetic and electrical fields so the Laplace/Poisson equations solutions represent the electrical fields alone. If there are no electical static charges, and the source exciting the electrical filed comes only from the boundary conditions, the laplace equation is used. Laplace Equation is a private case of Poisson's equation in which the electrical charges are zero. In both cases (Laplace/Poisson) the sollution is appropriate only for static, electrical fields, and boundary conditions should be applied in both cases of course.
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