With three of Maxwell’s equations covered there is just one more to go, but before we dig into it,
let’s get through some background information. For the vast majority of human history, the only known
sources of magnetic fields were certain iron ores and other materials that were magnetized in one
way or another. In 1820, French physicist Andre-Marie Ampere heard that in Denmark,
Hans Oersted had deflected a compass needle by passing an electric current nearby. Within a week,
Ampere had begun quantifying the relationship between electric currents and the magnetic field.
For many years, Ampere’s Law was known to apply only to static scenarios with steady currents.
Maxwell’s addition of a new term for changing electric flux in the 1850s allowed Ampere’s
law to apply to time-dependent situations. Here is the integral form of the Ampere-Maxwell
equation. Let’s unpack this a little bit before applying it. The left side represents the
circulation of the magnetic field along a closed path C. The right side is divided into two parts:
the electric current I, and the changing electric flux through a surface S, which is this whole term
over here. And we also have the permeability of free space, represented by mu nought. This
differs from the other term we are familiar with, epsilon, which was the permittivity of free space.
Permeability of free space deals specifically with magnetic fields.
The main idea of this equation is that an electric current or a changing electric flux
through a surface produces a circulating magnetic field around any path that bounds that surface.
This idea is rather similar to the other three equations. The way the magnetic field term is
surrounded by an integral and a dot product echoes the structure of Gauss’s law. The law itself is
almost a mirror of Faraday’s law. Instead of a changing magnetic field leading to a circulating
electric field, we see a changing electric field leading to a circulating magnetic field.
In many Ampere-Maxwell problems, the main consideration is the selection of
this path. Often called an ’Amperian Loop’, careful consideration that takes advantage
of any sym- metrical elements within the system can simplify the evaluation of
this equation and allow one to determine the generated magnetic field directly.
Generally, the equation is used to determine the circulation, or curl, of the magnetic field,
or the magnetic field directly from a current, meaning just the vector field, B.
It is rare to see it used in the other direction to find the electric current from the circulating
magnetic field, due to the dynamic nature of the two electrical terms on the right side.
Like the electric field in Gauss’s law, the magnetic field in Ampere-Maxwell is buried
inside an integral and coupled to a vector via a dot product. This further implies that a high
level of symmetry is required to determine the magnetic field from the electric current,
or a changing electric flux. The challenge arises in finding a way to
move the magnetic field term out of the integral. The differential form of the Ampere-Maxwell Law
is more often used in the opposite direction of the integral form, to find the electric current
density or change in electric flux from the circulating magnetic field. This form looks
like this, with the cross product, or curl of the magnetic field on the left, and then this other
term on the right. Mu nought is back again, which is the permeability of free space, times this
sum. J is the current density, and then we have epsilon nought, the permittivity of free space,
times the changing electric field with respect to time in the region in question. These generally
can’t be determined from the curl at the same time, they have wildly different influences on
the equation, and mathematically separating them is often impossible. However, one element of the
right side is usually purposefully eliminated from consideration by circumstance or boundary
conditions, to make things simpler mathematically. Note the differences between the integral form and
the differential form’s terms. The current density is listed, rather than the total enclosed current.
The electric field is directly referenced as a changing electric field, rather than
the indirect total flux through a surface. On top of this, the circulating magnetic field is
outside any integrals. However, to find either the changing electric field or the current density,
one must assume the current is steady or that there is no current running through
the given area. Otherwise, the two effects will be indistinguishable within the results.
That concludes an introduction to the Ampere-Maxwell equation,
and to Maxwell’s equations in general. In summary, each of these equations relates the properties
of an electromagnetic field in any classical situation. Whether that situation is actually
solvable without immense computational power is entirely due to how you approach the problem.
Scenarios that seem untenable can be made trivial through the appropriate application
of these equations. And with this basic understanding of Maxwell’s four Equations,
one can begin the long process of practicing their application through problem sets and
experimentation. Let’s now move on to some other topics in classical physics.