Laplace's equation

Laplace’s equation is a partial differential equation that governs several physical and theoretical phenomena. This includes electrostatic potential, heat transfer, fluid flow, wave propagation, complex analysis, differential geometry, and image and signal processing.

In ${\mathbb{R}}^3$, it’s given by:

$$\frac{{\partial }^{2}V}{\partial x^2}+\frac{{\partial }^{2}V}{\partial y^2}+\frac{{\partial }^{2}V}{\partial z^2}=0$$ $${\nabla }^{2}V=0$$

where ${\nabla }^2$ is the Laplacian operator. A function is harmonic if it satisfies Laplace’s equation.

In electromagnetism, Laplace’s equation is a special case of Poisson’s equation where the medium under consideration contains no charges, i.e., ${\rho }_v=0$.

Equivalent formulas

In cylindrical coordinates:

$${\nabla }^{2}V=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial V}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}V}{\partial {\phi }^2}+\frac{{\partial }^{2}V}{\partial z^2}$$

In spherical coordinates:

$${\nabla }^{2}V=\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial V}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial V}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}V}{\partial {\phi }^2}$$

Computations

For example, take a coaxial cable (i.e., two infinitely long cylinders, one nested within the other). Assume some non-uniform dielectric (i.e., it won’t reduce down right away; the example below is just an example).

The Poisson equation will expand out to:

$$\frac{1}{r}\frac{\partial }{\partial r}(r({ϵ}_a+{ϵ}_b{\cos }^2\phi )\frac{dv}{dr})=0$$

The term within the outside derivative is set to a term $C_1$.

$$C_1=r({ϵ}_a+{ϵ}_b{\cos }^2\phi )\frac{dv}{dr}$$

Rearrange as needed, and integrate to find $v$. This will net you an expression in terms of two constants. We can then apply the BVP to find the values of both constants.

Then we can take the gradient of the potential to find the electric field. Our final step: does this make physical intuitive sense? If it does, we’ve done something right.

Properties

There are two big properties that hold in the one, two, and three dimensional case:¹

• Laplace’s equation acts as an averaging function; i.e., the value of $V$ at some point is the average of those around the point.
• The equation has no local maxima or minima, and extreme values of $V$ must occur at the end points. This is a result of the above, since $V$ would be greater at a local maxima than on either side, and could not serve to be the average.
• The result of the above two properties mean that solutions ($V$; which are harmonic functions) show smooth, wave-like behaviour.
• Laplace’s equation follows superposition as it’s a linear PDE. The sum of any two harmonic functions is also a harmonic function.

Uniqueness

The first and second uniqueness theorems (given by the Dirichlet and Neumann boundary conditions, respectively) state that:

• The solution to Laplace’s equation in a volume $v$ is unique if $V$ is specified on the boundaries of $v$ (a surface $S$).
• In a volume $v$ surrounded by conductors containing a charge density $\rho$, the electric field is unique if the total charge on each conductor is given.

What are the implications of this? Let’s say we have a problem with a certain set of conditions, and we formulate a different and easier to solve problem with the same conditions. The solution to the different problem is a valid solution for the first problem, because of the Dirichlet condition.¹

If it satisfies Poisson’s equation in the region of interest, and assumes the correct value at the boundaries, then it must be right.

This is the foundation for the method of images.

Footnotes

1. From Introduction to Electrodynamics, by David Griffiths ↩ ↩²