4.10: The Laplacian Operator

The Laplacian $\nabla^2 f$ of a field $f({\bf r})$ is the divergence of the gradient of that field:

$$\nabla^2 f \triangleq \nabla\cdot\left(\nabla f\right) \label{m0099_eLaplaceDef}$$

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

$$\nabla^2 f= \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \label{m0099_eLaplaceScalar}$$

as can be readily verified by applying the definitions of gradient and divergence in Cartesian coordinates to Equation $\ref{m0099_eLaplaceDef}$.

The Laplacian relates the electric potential (i.e., $V$, units of V) to electric charge density (i.e., $\rho_v$, units of C/m$^3$). This relationship is known as Poisson’s Equation:

$$\nabla^2 V = - \frac{\rho_v}{\epsilon} \nonumber$$

where $\epsilon$ is the permittivity of the medium. The fact that $V$ is related to $\rho_v$ in this way should not be surprising, since electric field intensity $({\bf E}$, units of V/m) is proportional to the derivative of $V$ with respect to distance (via the gradient) and $\rho_v$ is proportional to the derivative of ${\bf E}$ with respect to distance (via the divergence).

The Laplacian operator can also be applied to vector fields; for example, Equation $\ref{m0099_eLaplaceScalar}$ is valid even if the scalar field “$f$” is replaced with a vector field. In the Cartesian coordinate system, the Laplacian of the vector field ${\bf A} = \hat{\bf x}A_x + \hat{\bf y}A_y + \hat{\bf z}A_z$ is

$$\nabla^2 {\bf A} = \hat{\bf x}\nabla^2 A_x + \hat{\bf y}\nabla^2 A_y + \hat{\bf z}\nabla^2 A_z \nonumber$$

An important application of the Laplacian operator of vector fields is the wave equation; e.g., the wave equation for ${\bf E}$ in a lossless and source-free region is

$$\nabla^2{\bf E} + \beta^2{\bf E} = 0 \nonumber$$

where $\beta$ 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:

$$\nabla^2 {\bf A} = \nabla\left(\nabla\cdot{\bf A}\right) - \nabla\times\left(\nabla\times{\bf A}\right) \nonumber$$

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