4.10: The Laplacian Operator
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The Laplacian ∇2f of a field f(r) is the divergence of the gradient of that field:
∇2f≜
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.