Laplace`s Equation in Electromagnetism

In electromagnetic theory, the electric field $\bf E$ is defined in terms of the electric potential $\phi$ by

$${\bf E} = -\nabla\phi .$$

For charge density $\rho$, one of Maxwell's equations states that $\bf E$ satisfies

$$\nabla\cdot{\bf E} = \rho / \epsilon_0 ,$$

so $\phi$ satisfies Poisson`s's equation:

$$\nabla^2\phi = \rho / \epsilon_0 .$$

In a charge-free region, $\rho=0$ and Poisson`s's equation becomes Laplace`s equation:

$$\nabla^2\phi = 0 .$$

In this course, we will confine our studies to this particularly simple, but very important, equation. In cartesian coordinates, it takes the form

$$\frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi... ...al y^2} + \frac{\partial^2\phi}{\partial z^2} + \cdots = 0 .$$

One particularly important property of Poisson`s's equation is that, if the potential $\phi$ is specified on the boundary of some region $R$, then that uniquely determines $\phi$ everywhere within $R$. Put another way, once we have obtained--by analytic or numerical means--a solution of Laplace`s equation that satisfies the appropriate boundary conditions, then we know we have the solution to the problem.