4.5: Gradient

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: $${\bf E}({\bf r}) = -\nabla V({\bf r}) \nonumber$$ or, with the understanding that we are interested in the gradient as a function of position ${\bf r}$, simply $${\bf E} = -\nabla V \nonumber$$

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.