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Poisson’s and Laplace’s Equations

In the earlier sections, we have determined the electric field Description: 37532.png in a region using Coulomb’s law or Gauss’ law when the charge distribution is specified in the region or using the relation Description: 25372.png when the potential V is specified throughout the region. However, in practical cases, neither the charge distribution nor the potential distribution is specified, and the electrostatic conditions (charge and potential) are specified only at some boundaries. These types of problems are known as electrostatic boundary value problems. For these types of problems, the field Description: 28278.png and the potential V are determined by using Poisson’s equation or Laplace’s equation.

For a linear material medium, Poisson’s and Laplace’s equations can easily be derived from Gauss’ law.
Description: 218795.png

This equation is known as Poisson’s equation which states that the potential distribution in a region depends on the local charge distribution.

In many boundary value problems, the charge distribution is involved on the surface of the conductors for which the free volume charge density is zero, i.e., ρ = 0. In that case, Poisson’s equation reduces to,
Description: 36368.png
This equation is known as Laplace’s equation.

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