• Electromagnetics I
  • Ch 4: Vector Analysis
  • Loc 4.5
  • Electromagnetics I
  • Ch 4
  • Loc 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

to the electric potential field

. 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 is the direction in which decreases most quickly, and
  • The scalar part of is the rate of change of in that direction. Note that this is also implied by the units, since has units of V whereas has units of V/m.

The gradient is the mathematical operation that relates the vector field

to the scalar field

and is indicated by the symbol “

” as follows:

(4.5.1)

or, with the understanding that we are interested in the gradient as a function of position

, simply

(4.5.2)

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

of some scalar field

. Furthermore, we find that other differential operators that are important in electromagnetics can be interpreted in terms of the gradient operator

. These include divergence (Section 4.6), curl (Section 4.8), and the Laplacian (Section 4.10).

In the Cartesian system:

(4.5.3)

Exercise

EXAMPLE 4.5.1: GRADIENT OF A RAMP FUNCTION.


Find the gradient of

(a “ramp” having slope

along the

direction).

Here,

and

. Therefore

. Note that

points in the direction in which

most rapidly increases, and has magnitude equal to the slope of

in that direction.

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

Additional Reading:

 

Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 CC BY-SA 4.0

 
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