Maxwell's Equations, Part VI

Maxwell's equations are expressed in the language of vector calculus, so a significant part of some previous installments in this series have been devoted to explaining vector calculus, not spectroscopy. In this installment, we will generalize our results from last time and consider how it applies to one of Maxwell's equations.
Apr 01, 2012
Volume 27, Issue 4

This is the sixth installment in a series devoted to explaining Maxwell's equations, the four mathematical statements upon which the classical theory of electromagnetic fields — and light — is based. Previous installments can be found on Spectroscopy's website ( Maxwell's equations are expressed in the language of vector calculus, so a significant part of some previous columns have been devoted to explaining vector calculus, not spectroscopy. In the last installment, we were developing the vector calculus tools to express Maxwell's equations. An understanding of the math makes it easier to understand the equations — that's why we're spending so much time on math, rather than spectroscopy. Apologies if this frightens or dismays you, but that's the way it goes. Once again, the numbering of the figures is intentional; it's meant to provide continuity throughout this entire series.

Where We Left Off

In our last installment, we left off with an expression for the infinitesimal work per unit area in a rectangular field in the (x,y) plane. The overall vector field is given by the equation

and we were assuming a closed path in just the aforementioned plane. We found that the infinitesimal work was

Thus, the closed loop in the (x,y) plane actually related to a vector in the z direction. The integral over the closed path is also referred to as the circulation of the vector field.

Now we are ready to generalize our result and consider how it applies to an equation — specifically, one of Maxwell's equations.

Introducing the Curl

For a vector function F = F x i + F y j + F z k, I will hereby define the function

as the one-dimensional curl of F. I designate it "one dimensional", possibly improperly, because the result is a vector in one dimension, in this case the z dimension. The analysis we performed in the earlier section — defining a closed path in a single plane and taking the limit of the path integral — can be performed for the (x,z) and (y,z) planes. When we do that, we get the following analogous results:

The combination of all three expressions gives us a general expression for the curl of F :

This expression allows us to determine

for any vector function F in any plane.

But what does the curl of a vector function mean? One way of thinking about it is that it is a variation in the vector function F that causes a rotational effect in a perpendicular axis. (Indeed, "curl F" is sometimes still designated "rot F," and a vector function whose curl equals zero [see Figure 38] is termed "irrotational.") Furthermore, a vector function with a nonzero curl can be thought of as curving around a particular axis, with that axis being normal to the plane of the curve. Thus, the rotating water in Figure 37 of our previous installment has a nonzero curl, while the linearly flowing water in Figure 38 has a zero curl. You may want to refer to the previous installment of this column to refresh your memory of what they look like (1).

A mnemonic (that is, a memory aid) for the general expression for curl F takes advantage of the structure of a 3 × 3 determinant:

Figure 39
Understand that curl F is not a determinant; a determinant is a number that is a characteristic of a square matrix of numerical values. However, the expression for curl F can be constructed by performing the same operations on the expressions as one would do with numbers to determine the value of a 3 × 3 determinant: constructing the diagonals and adding the right-downward diagonals and subtracting the left-upwards diagonals. If you have forgotten how, Figure 39 shows how to determine the expression for the curl.

The determinantal form of the curl can be expressed in terms of the del operator, ∇. Recall from part III of this series (2) that the del operator is

Also recall from vector calculus that the cross product of two vectors Ai A x + j A y + k A z and B defined analogously is written A × B and is given by the expression

By comparing this expression to the determinantal form of the curl, it should be easy to see that the curl of a vector function F can be written as

Like the fact that curl is not technically a determinant, it is technically not a cross product, as del is an operator, not a vector. The parallels, however, make it easy to gloss over this technicality and use the "del cross F" symbolism to represent the curl of a vector function.

Because the work integral over a closed path through an electrostatic field E is zero, it is a short, logical step to state that therefore

This is one more property of an electrostatic field: The field is not rotating about any point in space. Rather, an electrostatic field is purely radial, with all field "lines" going from the point in space straight to the electric charge.

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