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General Procedure for Solving Poisson's and Laplace's Equations

The following procedure may be followed for solving a boundary value problem using Poisson’s or Laplace’s equation.
  1. The equation is solved using either
    • Direct integration when V is a function of one variable
    • Separation of variables if V is a function of more than one variable
  2. The unknown integration constants are found by applying the boundary conditions so that the solution becomes unique.
  3. Having obtained V, the field is obtained from the relation Description: 34058.png
  4. The charge induced on a conductor is obtained from the relation Description: 30737.png where σ is the induced surface charge density, Description: 154714.png and where Description: 154705.png is the normal component of the field.
  5. The capacitance between two conductors is obtained from the relation Description: 154725.png

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