Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Fields change from point to point, and the laws of electromagnetics are statements about how they change. The rates of change of a function of three coordinates are its partial derivatives, and it is convenient to package the three first-order partial derivatives into a single object, the nabla (or del) operator:

\nabla = \frac{\partial}{\partial x} a_{x} + \frac{\partial}{\partial y} a_{y} + \frac{\partial}{\partial z} a_{z} . (2.2.1)

Nabla is a vector operator: it has the formal structure of a vector, with components that are first-order partial derivatives with respect to the space coordinates, but it is not a vector by itself: it does nothing until it acts on a function standing to its right. It can be applied to a scalar function or to a vector function, and, because it behaves like a vector, it can act in the three ways in which a vector can multiply something (Figure 2.2.1):

"times a scalar" $f$ : $\nabla f$ , the gradient, a vector,
"dot" a vector $A$ : $\nabla \cdot A$ , the divergence, a scalar, and
"cross" a vector $A$ : $\nabla \times A$ , the curl, a vector.

Combining the first two gives a fourth, second-order operator, the Laplacian $\nabla \cdot \nabla f = \nabla^{2} f$ . Because $\nabla$ contains derivatives, its position matters: $\nabla \cdot A$ is a scalar function, whereas $A \cdot \nabla = A_{x} \partial / \partial x + A_{y} \partial / \partial y + A_{z} \partial / \partial z$ is again an operator. Each derivative carries a factor $[1/m]$ , so $\nabla$ applied to a quantity measured in volts gives a result in $[V/m]$ .

What goes in and what comes out of the four differential operators built from \nabla. Keeping track of “scalar” and “vector” is the first check of any calculation: expressions such as the divergence or the curl of a scalar function have no meaning. — **Figure 2.2.1.** What goes in and what comes out of the four differential operators built from $\nabla$ . Keeping track of “scalar” and “vector” is the first check of any calculation: expressions such as the divergence or the curl of a scalar function have no meaning.

Quick check

Which of these expressions make sense? (i) $\nabla \cdot f$ , (ii) $\nabla \times (\nabla f)$ , (iii) $\nabla (\nabla \cdot A)$ , (iv) $\nabla \cdot (\nabla \cdot A)$ .

Follow the types of Figure 2.2.1. (i) No: divergence needs a vector, $f$ is a scalar. (ii) Yes: $\nabla f$ is a vector, and so is its curl (we shall find that it is always $0$ ). (iii) Yes: $\nabla \cdot A$ is a scalar, and its gradient is a vector. (iv) No: $\nabla \cdot A$ is a scalar, which has no divergence.

2.2.1 The nabla operator