So far in this series, we have covered the major topics in classical physics at a level of depth
appropriate for a high school physics course. Now it is time to tackle select concepts like
electricity and magnetism with significantly more depth, so that we can gain a more
sophisticated understanding of the universe. We will begin this next phase of the series by
examining Maxwell’s equations. We have mentioned these in passing, but we did not examine them with
any rigor. This is because a proper treatment of these equations requires fairly advanced
mathematics, even beyond basic calculus. Those who wish to move forward with this series will
need to understand vector fields, divergence and curl, and other topics in linear algebra.
Fortunately I have covered all of these in my mathematics series, so if you feel that
you want to move forward with physics but you don’t have the necessary background in math,
head over there to get some prerequisites covered before watching this tutorial. If
you’re all set with the math, let’s get started. Maxwell’s equations for electromagnetism elegantly
correct and combine almost a hundred years of work by earlier well-known scientists,
those being Coulomb, Gauss, Ampere, and Faraday. Formulated almost 200 years ago, they are still
the simplest way to accurately represent the fundamental laws of the electromagnetic
field and its carrier particle, the photon. Each of these four equations has two forms:
a differential form and an integral form. We are going to examine Maxwell’s four equations
one at a time over the next four tutorials. The first one is Gauss’s law for the electric field,
so let’s start with the foundational concepts behind the derivation of this law.
As we mentioned, this law has both differential and integral forms. These two forms are useful
in entirely separate situations. The differential form states that the divergence of the electric
field in a given region is related to the charge density within that region. Remember that
divergence means we are finding this dot product, where the del operator is a vector comprised of
the relevant partial differential operators, and that the divergence of a vector field,
like this electric field, gives us a scalar field. This form is used to determine the divergence
of an electric field at a particular point if the volumetric charge density, rho, is known,
meaning the quantity of charge per unit volume, measured in coulombs per cubic meter. Epsilon
nought is a constant representing the permittivity of free space, or the capability of a vacuum to
permit electric fields. Conversely, if the spatial variation of the vector electric field is known,
the differential form allows us to find the volumetric charge density at a particular point.
The integral form can be derived directly from the differential form by applying the divergence
theorem, which we went over in one of the mathematics tutorials. Let’s quickly run through
the derivation together. First we will take the differential form and integrate both sides with
respect to dV, which is the differential volume, essentially the same as dx dy dz in Cartesian
space. Now, the left side of this equation resembles one side of the divergence theorem.
This theorem says that the triple integral of the divergence of a vector field, which is what
is represented here given the del operator and the fact that we are integrating with respect to dV,
is equal to this surface integral. So we can change what we have to this surface integral,
and for our purposes we can simply express it in this way, because the electric field is the
force in this situation, n hat is the normal unit vector as we would expect, and then da
is the integrand for this surface integral that will allow us to express it as a single integral.
Then the final step is to reduce the integral on the right side. Knowing that epsilon nought
is a constant, and the charge density rho taken over the entire enclosed volume is simply the
enclosed charge q, that will leave us with this. And there we have the integral form of this law.
This form states that the flux passing through a closed surface is proportional to the charge
enclosed therein. The flux of an electric field can be thought of as analogous to the
“amount of fluid flow” through an imaginary flat surface. This equation can be used to
determine the flux of an electric field from the enclosed charge, or the total enclosed charge
from the flux. Unlike the differential form, the integral form acts over a surface area.
Now, each of Maxwell’s Equations describes the electromagnetic field in a different way. They
are tailored to meet the requirements of a certain problem set in the real world.
Gauss’s Law for the electric field excels in the realm of electrostatics, where all charges or
charged objects are stationary and the boundary conditions are well known. For this reason,
it is often the simplest starting point. The practice problems associated with this
equation usually involve simpler shapes, like spheres, infinite sheets, and infinite cylinders.
Let’s look at the two forms of this law again. In both forms, there is the constant we described
earlier, the permittivity of free space. This constant is only valid for situations where all
currents, both bound and free, are accounted for. In a more approachable situation than
the vast emptiness of deep space or a large laboratory vacuum tube, this value is set
by the material or materials that the electric field in question interacts with. In other words,
the permittivity differs from one material to another, with free space acting as the baseline,
represented by epsilon nought, whereas epsilon without the nought will represent a permittivity
constant for a specific material, and we can define the vector field D in this manner,
as the electric field times permittivity. In such a scenario, Gauss’ Law can be expressed
in this manner, where the permittivity constant is not shown because it is accounted for in D.
The same goes for the corresponding integral form. Now let’s apply this understanding to solve a
problem. Consider the integral form of Gauss’s Law. Finding the electric field from a given
charge distribution may seem a daunting, if not impossible task. From the dot product,
only the normal component emerges, leaving the electric field term trapped within the integral.
There exists a very limited set of scenarios, all characterized by a high degree of symmetry,
that allow us to both remove the electric field from the integral and simplify the dot
product into simple algebraic multiplication. Thus, with Gauss’ Law for the electric field,
the electric field must be perpendicular or parallel to the surface, and the electric
field must be a constant or zero over the surface. Let’s solve a problem that meets these conditions.
Suppose there is a small charged sphere in space with a uniform volume charge density, rho, and
radius, a. We need to find the electric field at a distance r from the center of the charged sphere.
To do this, we must create a special Gaussian surface around the charged sphere. This surface
has to meet the two criteria we mentioned before, and a sphere of radius r will do it. The electric
field will be perpendicular to the surface, and the electric field will be constant over
the surface because the charge distribution inside the Gaussian surface is spherically symmetrical.
We will then manipulate Gauss’s law according to this particular situation. We can pull
the electric field out of the integral because it’s constant over the surface,
and we can pull the normal vector out of the integral because it is perpendicular to the
surface of the Gaussian sphere and therefore reduces to one. Then we must integrate da.
This represents the surface area of the Gaussian sphere, and the formula for the surface area of
a sphere is four pi r squared, given that we have selected r as the radius for the Gaussian sphere.
Then we can simply divide both sides by four pi r squared to solve for the electric field.
Thus, the electric field will be equal to the total charge enclosed, divided by the surface area
of the sphere and the permittivity of free space. Let us notice that this is rather similar to
Coulomb’s Law, which we introduced in a previous tutorial. In fact, if we acknowledge this value
for k and make the substitution in Coulomb’s Law, we see that they are nearly identical.
We are not quite finished, however. The enclosed charge varies depending
on the size of the chosen Gaussian sphere. Let us examine three scenarios. For the first
scenario, if it is smaller than the charged sphere, meaning r is less than a, then we can
actually say that a = r, because we will only be considering the charge density within the
Gaussian. This means that q equals rho times the volume of the Gaussian, or four thirds pi r cubed.
We can plug this into our previous expression, and we will see that four will cancel, pi will cancel,
and two of the r terms will cancel. The electric field would then simplify to this expression.
For the second scenario, if we were to expand the Gaussian sphere so that it was exactly the
same size as the charged sphere, such that r = a, we would maximize the electric field as follows,
by simply swapping r for a. For the third and final scenario, let’s move our Gaussian sphere
back outside the charged sphere, such that r is greater than a. The enclosed charge would now
be a constant, but the distance r would continue to increase. Thus we have this expression for q,
with a in it, and that leaves us with this, which is essentially the same as the previous situation,
except that the r’s don’t cancel and we are left instead with a cubed on top.
Any of these expressions we have just derived can then be easily
evaluated to calculate the electric field. And that gives us a picture of what we can
calculate with this law. On its own, Gauss’s Law is not capable of interpreting the changing,
shifting world of electrodynamics that we inhabit. It is an incomplete picture, tailored to a static,
tightly controlled scenario. In the next tutorial, we will cover the magnetic side of Gauss’s
Law. This second of Maxwell’s equations will complete the static picture of electromagnetism,
and lay the foundation for interpreting the dynamics of our ever-changing world.