In the previous tutorial we introduced Maxwell’s equations, and we examined the first of them in
some depth, Gauss’s law for the electric field. In this tutorial we will cover the second of
Maxwell’s equations, which is Gauss’s Law for the magnetic field. Unlike with the electric field,
magnetic ”charge” does not exist in isolation. Magnetic poles must always exist in apparent
pairs. This is mathematically expressed within Gauss’s Law, and can be physically represented
by observing the perpendicular nature of the components of the electromagnetic field,
shown here with E representing the electric field propagating on the y-axis,
and B representing the magnetic field propagating on the z-axis.
We were already aware of this relationship conceptually, so now let’s look at the math.
This magnetic half of Gauss’s Law arises from the idea that a moving electric charge
produces a magnetic field. Just like the first equation, there are two forms of this equation,
a differential form and an integral form. The integral form of Gauss’s Law for the magnetic
field states that the total magnetic flux passing through any closed surface is zero.
Both forms arise directly from the idea that there are no isolated magnetic poles in nature,
unlike electric charge. The main idea of this law can only be frugally applied in practice. Problems
with complex surfaces can be solved completely if the flux through one part of the surface is
already known. Rarely does a problem exist that can be solved directly with the integral form. It
is more appropriately used as a boundary condition to limit the number of possible solutions. The
differential form, on the other hand, states that the divergence of the magnetic field in a given
region is equal to zero. We can easily derive the integral form from the differential form
in precisely the same manner as we did for the first of Maxwell’s equations. Since the method is
identical we will just quickly summarize it here. We can integrate both sides with respect to dV,
which means the right side will remain zero, and then we apply the divergence theorem, allowing
us to replace this expression with this surface integral. And there we have the integral form.
As a quick reminder, looking again at the differential form, the divergence of a field is
the tendency of that field to ”flow” more strongly away from a point than towards it. A divergence of
zero means a ”flat” field. This is an important property, and it reaffirms the idea that a
magnetic field does no work. There is no ”magnetic charge density” because the magnetic field does
not originate from a point source, but rather from a moving electric charge, which is a vector.
These two equations are difficult to use directly in practice.
More often, these equations are used to resolve a difficult or complex situation arising from other,
more common equations, such as the Biot-Savart Law. For example, if you know the magnetic field
within a part of a complex shape, Gauss’s Law can be used to determine the remaining field.
On principle, Gauss’s Law for the magnetic field displays the simplest facts about the
magnetic field. A great representation of the magnetic flux is the idea of magnetic field
lines. Unlike electric field lines, magnetic field lines do not terminate. They are complete loops
that pass through the north and south poles of the magnetic field. No matter what closed surface
one draws around these lines, there will always be an equal number of lines going out and coming
in. This satisfies Gauss’s Law: the total flux is zero. In the next two tutorials after this one,
the relationship between the electric field and magnetic field will be explored further.
Before we get there, let’s take a look at a problem using the results of the two forms
of Gauss’s law. Given the following expression for a vector field, let’s determine whether the
field could be a magnetic field. Let’s think for a moment about the best way to do this. Remember,
Gauss’s Law tells us that the divergence of the magnetic field is zero. Therefore, the simplest
thing to do is to take the divergence of the given expression and see if it goes to zero.
So we can apply the del operator here on the left, and remember that means
we are now taking a dot product with a vector containing the partial differential operators,
so we end up with d dx applied to the first term, plus d dy applied to the second term,
and we have removed the unit vectors for simplicity. For this first term, taking
the derivative with respect to x requires that we use the chain rule. The derivative of cosine
is negative sine, and then we must multiply by the derivative of what’s inside here, so we get an
additional b term, so that’s negative ab sine bx. For the second term, taking the derivative with
respect to y, everything but y is a constant, so we just lose y and keep the rest, or ab sine bx.
Adding these together, we do indeed get zero, which indicates that A could be a magnetic field.
With that we complete our overview of Gauss’s Law, which represents two of Maxwell’s equations.
The final two of Maxwell’s equations deal with the dynamics of electromagnetism, rather than
the static picture that Gauss’ Law operates in. So let’s move forward and examine Faraday’s Law next.