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

Spectroscopy

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
x,y) plane. The overall vector field is given by the equation
Thus, the closed loop in the ( Now we are ready to generalize our result and consider how it applies to an equation — specifically, one of Maxwell's equations.
F = F
_{x}
i + F
_{y }
j + F
_{z}
k, I will hereby define the function
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:
curl of
:
F
for any vector function 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 takes advantage of the structure of a 3 × 3 determinant:
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.
A ≡ i
A
_{ x } + j
A
_{ y } + k
A
_{ z } and B defined analogously is written A × B and is given by the expression
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
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. |