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15.3: Poisson's and Laplace's Equations

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  • Equation 15.2.4 can be written \( \bf{\nabla \cdot E} = \rho/ \epsilon\), where \(\epsilon\) is the permittivity. But \(\bf{E}\) is minus the potential gradient; i.e. \(\bf{E} = -\nabla V\). Therefore,

    \[ \nabla^2 V = \dfrac{\rho}{\epsilon} \tag{15.3.1} \label{15.3.1}\]

    This is Poisson's equation. At a point in space where the charge density is zero, it becomes

    \[ \nabla^2 V = 0 \tag{15.3.2} \label{15.3.2}\]

    which is generally known as Laplace's equation. Thus, regardless of how many charged bodies there may be an a place of interest, and regardless of their shape or size, the potential at any point can be calculated from Poisson's or Laplace's equations. Courses in differential equations commonly discuss how to solve these equations for a variety of boundary conditions – by which is meant the size, shape and location of the various charged bodies and the charge carried by each.

    It perhaps just needs to be emphasized that Poisson’s and Laplace’s equations apply only for static fields.