The Laplacian Operator

The Laplacian

of a field

is the divergence of the gradient of that field:

Note that the Laplacian is essentially a definition of the second derivative with respect to the three spatial dimensions. For example, in Cartesian coordinates,

as can be readily verified by applying the definitions of gradient and divergence in Cartesian coordinates to Equation 4.10.1.

The Laplacian relates the electric potential (i.e.,

, units of V) to electric charge density (i.e.,

, units of C/m

). This relationship is known as Poisson’s Equation (Section 5.15):

where

is the permittivity of the medium. The fact that

is related to

in this way should not be surprising, since electric field intensity (

, units of V/m) is proportional to the derivative of

with respect to distance (via the gradient) and

is proportional to the derivative of

with respect to distance (via the divergence).

The Laplacian operator can also be applied to vector fields; for example, Equation 4.10.2 is valid even if the scalar field “

” is replaced with a vector field. In the Cartesian coordinate system, the Laplacian of the vector field

is

An important application of the Laplacian operator of vector fields is the wave equation; e.g., the wave equation for

in a lossless and source-free region is (Section 9.2)

where

is the phase propagation constant.

It is sometimes useful to know that the Laplacian of a vector field can be expressed in terms of the gradient, divergence, and curl as follows:

The Laplacian operator in the cylindrical and spherical coordinate systems is given in Appendix B2.

• “Laplace operator” on Wikipedia.