Notice the similarity of the law of motion (27) to the
law of motion for sound (7) and the corresponding
equation string. We find the spatial derivative of one quantity to be
related directly to the time derivative of another. It thus appears
that 
plays the role of 
, some sort of driving force, and 
plays the
role of 
, some sort of momentum.
Carrying the analogy further, we thus expect that 
plays the
role of B, some sort of stiffness, and that 
plays the
role of 
, some sort of inertia. If this analogy is to be
complete, then we need also that 
play the role of 
, the spatial derivative of some quantity, and that 
play the role of 
, the time derivative of the
same quantity.
In general, it is not always possible to define two
quantities like and to arise as two different derivatives of a
single quantity
.
What allows us to do so in this case, and thus make
the analogy complete, is Faraday's law (30). It
is a theorem of partial differential calculus that the mixed partial
derivatives of a single quantity ( 
and 
) are equal. Thus, if we
define to be the x derivative of some quantity and to
the time derivative of some quantity, then we must have that the t
derivative of equal the x derivative of . A further
theorem states that so long as these latter two derivatives indeed are
equal (as Faraday's law ensures for us!), then we indeed always can
define a single quantity whose derivatives yield the original two
quantities. In the present case, this
particular quantity has a name
in advanced courses in electromagnetism, the vector potential 
.
In such courses, you will learn that in
fact
and thus what plays the role of s is exactly the quantity
.
(The minus signs are a matter of convention.) As a final remark, if we
now insert the definitions (31), which Faraday's law
allows us to make, into Ampere's law (27), we could
derive the wave equation,
and immediately obtain the wave speed for electromagnetic waves,
. As the vector potential is not
part of the official material for this course, we take a somewhat more
roundabout approach to deriving the wave equation in the next section.
