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# 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:

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

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

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

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

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

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

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