The gradient operator is an important and useful tool in electromagnetic theory. Here’s the main idea:

The gradient of a scalar field is a vector that points in the direction in which the field is most rapidly increasing, with the scalar part equal to the rate of change.

A particularly important application of the gradient is that it relates the electric field intensity $${\bf E}({\bf r})$$ to the electric potential field $$V({\bf r})$$. This is apparent from a review of Section 2.2; see in particular, the battery-charged capacitor example. In that example, it is demonstrated that:

• The direction of $${\bf E}({\bf r})$$ is the direction in which $$V({\bf r})$$ decreases most quickly, and
• The scalar part of $${\bf E}({\bf r})$$ is the rate of change of $$V({\bf r})$$ in that direction. Note that this is also implied by the units, since $$V({\bf r})$$ has units of V whereas $${\bf E}({\bf r})$$ has units of V/m.

The gradient is the mathematical operation that relates the vector field $${\bf E}({\bf r})$$ to the scalar field $$V({\bf r})$$ and is indicated by the symbol “$$\nabla$$” as follows: ${\bf E}({\bf r}) = -\nabla V({\bf r})$ or, with the understanding that we are interested in the gradient as a function of position $${\bf r}$$, simply ${\bf E} = -\nabla V$

At this point we should note that the gradient is a very general concept, and that we have merely identified one application of the gradient above. In electromagnetics there are many situations in which we seek the gradient $$\nabla f$$ of some scalar field $$f({\bf r})$$. Furthermore, we find that other differential operators that are important in electromagnetics can be interpreted in terms of the gradient operator $$\nabla$$. These include divergence (Section 4.6), curl (Section 4.8), and the Laplacian (Section 4.10).

In the Cartesian system:

$\nabla f = \hat{\bf x}\frac{\partial f}{\partial x} + \hat{\bf y}\frac{\partial f}{\partial y} + \hat{\bf z}\frac{\partial f}{\partial z} \label{eDelCar}$

Example $$\PageIndex{1}$$: Gradient of a ramp function.

Find the gradient of $$f=ax$$ (a “ramp” having slope $$a$$ along the $$x$$ direction).

Solution

Here, $$\partial f/\partial x = a$$ and $$\partial f/\partial y = \partial f/\partial z = 0$$. Therefore $$\nabla f = \hat{\bf x}a$$. Note that $$\nabla f$$ points in the direction in which $$f$$ most rapidly increases, and has magnitude equal to the slope of $$f$$ in that direction.

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