Maxwell's Equations

Section 10.1 Maxwell's Equations

Subsection 10.1.1 19th Century Electromagnetic Observations

In this section, I'm presenting a brief version of background to vector calculus. All the major tools defined in this course (vector fields, curl, gradient, line integrals, flux integrals, Gauss-Green-Stokes) are 19th century mathematics. They were invented very specifically to describe electricity and magnetism. Without getting to deep into the physics, I want to show you how the physical property of electricity and magnetism required these tools how these tools are precisely what was needed for these problems.

I want to start with a summary of the experimental results that inspired this mathematics. This list below comes from people experimenting in laboratories and workships. Once it was understood how electricity could be generated and stored, people were playing around with various machines to see how electricity behaved. Magnets, of course, have been used since antinquity, but new methods of understanding how magnets worked were being developed at the same time. I'll start with observations of electricty and magnetism as two separate things. (Note that all the sign conventions are arbitrary. What observers decided was ‘positivie’ or ‘negative’ is just a choice of conventions. The opposite choice could have been made and also would have produced consistent mathematical models.)

Electric charge exists. There is such a thing as a electrically charged particle. Moreover, charge comes in two types: positive and negative. Charges of the same type repel each other and charges of opposite type attract each other.
Electric charges create electric fields. An electric field is a field of force in the volume surrounding a charged object. It acts on other charged object, creating an attractive force on objects of opposite charge and a repulsive force on object of the same chare.
Electric charge can move though a medium (usually metalic). Such a movement is called current. Typically, current refers to the flow of negative charges.
Unlike electric charge, isolated magnetic charge is impossible. There is no such thing as an isolated positive or negative magnetic charge (which likewise implies that there is no such things as magnetic current). Instead, magnetic charge only appears in dipoles: objects that have a positive pole and a negative pole.
Magnetic dipoles create a magnetic field. A magnetic field is a field of force in the volume surrounding a magnetic dipoles which acts on other magnetic dipoles. The field pulls the negative end of other dipoles towards the positive end of the original dipole, and likewise for the pulling the positive end of other dipoles towards the negative ned of the original dipole.

That summarizes the observations about electricity and magnetism by themselves. But perhaps the most important observation was the fact that electricity and magnetism interact. Moreover, the interact in some very strange and unexpected ways. Here is a list of various interactions that were observed.

A current (moving electric charge) through a wire creates a magnetic field. The direct of this field is circular, moving around the wire. The direction of the current and the direction of the rotation agree by a right-hand-rule. (The right-hand-rule is a result of the various sign conventions; different choices of positive and negative charge, or positive and negative ends of a magnet, would have lead instead to a left-hand-rule.)
As an application of the previous point, if the wire is wrapped into a solenoid (a ring formed of coiled wire, essentially), it creates a magnetic field inside the solenoid. The direction of this magnetic field is point through the ring (again, following a right-hand-rule) for the direction of the field and the direction of rotation of the current in the ring.
A changing magnetic field (produced, say, by spinning a magnetc and thus moving the resulting magnetic field) produces an electric field and thus an electric force on nearby electrically charged particles.
Magnetic fields act only on moving charged particles and induce torque to change the direction of movement.

The key piece in all these interactions is change or movement. A static electric field doesn't do anything to a stationy magnets. Likewise, a stationary magnets doesn't have an effect on a stational electrically charge particular. It's only when something is changing (the magnetic field is changing, the charge is moving in a curren, etc) that the interaction happens. The interactions are all dynamic.

Subsection 10.1.2 Mathematics Formalism for Electromagnetism

Now I'm going to try to take the observations from the previous list and describe them in a mathematical model. I'm going to use all the tools of this course, but remember, when this was first done, the mathematics tools had to be invented. First, let me describe the model for static situation.

Isolated electric charges exit. If there is a charge at a point, I can just attach a value \(q\) to that point. \(q\) is the conventional letter for charge and has units of coulombs. In the SI system, coulombs are a derived unit, since it was decided that current should be the fundamental unit. A coulomb is the same as an amp-second, written \(A\cdot s\text{.}\) In addition to point charges, often there is a distribution of charge in a region. This can be described as a charge density, which is a scalar field \(\rho(x,y,z)\text{.}\) Charge density has units of coulombs per cubic metre, \(\frac{C}{m^3}\) or amp-seconds per cubic metre, \(\frac{A \cdot s}{m^3}\text{.}\)
Electric charge (either a point charge or a distribution of charge in a region) creates a field \(E(x,y,z)\) of force per unit charge. The field acts on other potential charges in the region around the existing charge. The field \(E\) has unit of volts per meter. Again, in the SI system, the volt is a derived unit and the base units for electric fields are \(\frac{kg \cdot m}{s^3 \cdot A}\text{.}\)
In addition to charge, the flow of charge in current exists. This is usually described by a current density \(J\) which describe current flowing through a cross-sectional area. \(J\) has units of coulombs per second-square-metre. In SI units, \(J\) has units of \(\frac{A}{m^2}\text{.}\)
There is a universal constant which determine the strength of interaction due to electric fields. This constant is called the permittivity of the vacuum and has the value \(\epsilon_0 = 8.85 \times 10^{-12}\) with units \(\frac{s^4 \cdot A^2}{m^3 \cdot kg}\text{.}\)
If a charge density \(\rho\) is creating the electric field \(E\text{,}\) the relationship between the charge density and the field is given by the divergence differential operator: \(\nabla \cdot E = \frac{\rho}{\epsilon_0}\text{.}\)
Isoated magnetic charge does not exist, so not symbols or units are assigned to it.
A magnetic dipole creates a force \(B\text{.}\) Following the observations, this force acts on a moving electrically charged particle by changing its direction of movement. Therefore, it is natural to describe this field as thee force per charge-velocity The unit of \(B\) the telsa, which in SI base units is \(\frac{kg}{A\cdot s^2}\text{.}\)
There is a universial constant that describe the strengh of magnetic field interactions. This constant is called permeability of the vacuum and has value \(\mu_0 = 4\pi \times 10^{-7}\) with units \(\frac{m \cdot kg}{s^2 \cdot A^2}\text{.}\) (This universal constant is not independent of other constants; it is, in fact, a derived value. This explains why it has a value involving \(4\pi\) instead of just an arbitrary decimal.)
Since there are no monopoles, all magnetic charged is balanced. This is reflected in the mathematical by the equation \(\nabla \times B = 0\text{.}\)

Now I can move on to the dynamic interactions between electricity and magnetism. I'll start with the effect of statis fields on a charged particle. Both static situations (electric and magnetic) produce fields and forces on charged particles. The action on charged particles by both \(E\) and \(B\) is summarized in an equation called the Lorentz force law. If \(q\) is a charged particle travelling with velocity \(v\text{,}\) then the force caused by electric and magnetic fields on the charged particular satisfies this equation.

\begin{equation*} F = Eq + q (v \times B) \end{equation*}

Now I need the model to consider changinge electric and magnetic fields. A key part of the observation what that changes in electrical fields produce magnetic fields and vice-versa. How is this described, mathematically?

A changing \(E\) fields induces a \(B\) field and vice-versa.
In addition, a current induces a \(B\) field.
Magnetic current is impossible, so it cannot induce a \(E\) field.
The induced \(E\) field due to a changing \(B\) field satisfies a differential equation.

\begin{equation*} \nabla \times E = \frac{-\del B}{\del t} \end{equation*}
The induced \(B\) field due to a changing electric field \(E\) and a current density \(J\) also satisfies a differential equation.

\begin{equation*} \nabla \times B = \mu_0 J + \mu_0 \epsilon_0 \frac{\del E}{\del t} \end{equation*}

In this mathematical object, I have written down four differential equations. This is the first presentation of Maxwell's equations. (This version is the presentation via differential operators; a second presentation will follow below). There are four equations: two for statics, which calculate divergence of the fields, and two for dynmanics, which calculate curl of the fields. Without fields, curl and divergence, such differential equations are impossible to state; therefore, the concepts of vector fields, curl and divergence were invented in order to be able to state the fundamental equations that govern electricity and magnetism. To determine the time behaviour of an electromagnetic system means solving these four differential equations for whatever initial conditions describe the system. (This is, in general, a very difficult problem).

\begin{align*} \nabla \cdot B \amp = 0\\ \nabla \cdot E \amp = \frac{\rho}{\epsilon_0}\\ \nabla \times B \amp = \mu_0 J + \mu_0 \epsilon_0 \frac{\del E}{ \del t}\\ \nabla \times E \amp = \frac{-\del B}{\del t} \end{align*}

Subsection 10.1.3 Gauss-Green-Stokes and Maxwell's Equations

Some of the 19th century electro-magnetic observations directly measured charge, field and current, leading to the equations in the previous section. However, often the fields are difficult to measure directly; instead, their flux through various surfaces can be measured in the laboratory. Flux essentially measure the fields strength via particular apparatus in a laboratory. To that end, there is a second presentation of Maxwell's equations using the flux of fields, not just the fields themselves.

What is the relationship between the two presentations? How do I go from the equations above, with curls and divergences, to equations of flux? I apply Gauss' and Stokes' theorem to the first presentation of Maxwell's equations. This is why Gauss' and Stokes' theorems were discovered and what made them so fundamental to the discipline. They transform the equation of fields into equations of flux which can be measure more reasonable in laboratory situations.

Let \(D\) be a solid region of space and \(\sigma\) a parametric surface (not necessarily the boundary of \(D\)). Before stating the new version of Maxwell's equations, I need some new notations.

\(Q_D\) is the total charge in the solid region \(D\text{.}\)
\(I_\sigma\) is the total current flowing through a surface.
\(\Phi_{E,\sigma}\) is the flux of \(E\) through \(\sigma\text{.}\)
\(\Phi_{B,\sigma}\) is the flux of \(B\) through \(\sigma\text{.}\)

Now, if \(\sigma\) is the bounding surface for the region \(D\text{,}\) then I can apply Gauss' theorem to the field \(E\text{.}\)

\begin{equation*} \Phi_{E,\sigma} = \int_\sigma E \cdot dA = \int_D \nabla \cdot E dV = \int_D \frac{\rho}{\epsilon_0} dV = \frac{Q_D}{\epsilon_0} \end{equation*}

I can do the same for \(B\text{,}\) remembering that \(\nabla \cdot B = 0\text{.}\)

\begin{equation*} \Phi_{B,\sigma} = \int_\sigma B \cdot dA = \int_D \nabla \cdot B dV = \int_D 0 dV = 0 \end{equation*}

Now let \(\sigma\) be any surface (not the boundary of \(D\)) and let \(\gamma\) be its boundary. I can think of \(\gamma\) as a loop of wire, and \(\sigma\) the surface inside the loop. The line integral of \(E\) along the wire is the work done on by the field moving electrons along the wire. I can use Stokes' theorem and the firsr presentation of Maxwell's equation to calculate the line integral.

\begin{equation*} \int_\gamma E \cdot ds = \int_{\sigma} (\nabla \times E) \cdot dA = - \int_\sigma \frac{\del B}{\del t} \cdot dA = -\frac{\del}{\del t} \Phi_{B,\sigma} \end{equation*}

Thinking the other way around, this calculates the flux through the wire of the magnetic field induced by the current. If the loop is now a series of loops in a solenoid, the flux adds up, so more loops gives move surface area, hence more flux, hence a greater induced magnetic field. A transformer is formed from two solenoids, where a current in one solenoid induced a magnetic fields which, in turn, induces a new current in the other solenoid. In such a transformer, the designer can adjusting the ingoing and outgoing current by having different numbers of coils on each solenoid.

For magnetic fields, I can do the same Stokes' theorem calculation.

\begin{equation*} \int_\gamma B \cdot ds = \int_{\sigma} (\nabla \times B) \cdot dA = \int_\sigma \mu_0 J \cdot dA + \int_\sigma \mu_0 \epsilon_0 \frac{\del E}{\del t} = \mu_0 I_\sigma + \mu_0 \epsilon_0 \frac{\del}{\del t} \Phi_{E, \sigma} \end{equation*}

Now I can put all these equation together into the second presentation of Maxwell's equations. In the first two equations, \(\sigma\) is the closed boundary of \(D\text{.}\) In the second two equations, \(\sigma\) is an open surface with boundary \(\gamma\text{.}\)

\begin{align*} \int_\sigma E \cdot dA \amp = \frac{Q_D}{\epsilon_0}\\ \int_\sigma B \cdot dA \amp = 0\\ \int_\gamma E \cdot ds \amp = \frac{-\del}{\del t} \Phi_{B,\sigma}\\ \int_\gamma B \cdot ds \amp = \mu_0 I_{\sigma} + \mu_0 \epsilon_0 \frac{\del}{\del t} \Phi_{E,\sigma} \end{align*}

Subsection 10.1.4 Maxwell's Equations in a Vacuum

As I said above, actually solving Maxwell's equations is very difficult. This is particularly true since all of the physical condition of the situation are build into the initial conditions of the differential equations, which can be very complicated. However, I could choose the easiest initial condition possible: none at all.

This train of through leads to a very strange question: what happens if I consider Maxwell's equation in a vacuum with no external influence? There is nothing there, no charge, no current, no wires. However, the existence of electrical and magnetic fields in a vacuum is possible, so the differential equations still might have something to say. I'll work with the first presentation of Maxwell's equations here. I'll see all the charge and current terms to zero, since there are no charge particles, no currents, and no magnets in a vacuum. Here are the resulting differential equations.

\begin{align*} \nabla \cdot B \amp = 0\\ \nabla \cdot E \amp = 0\\ \nabla \times B \amp = \mu_0 \epsilon_0 \frac{\del E}{\del t}\\ \nabla \times E \amp = \frac{-\del B}{\del t} \end{align*}

This is a difficult system of differential equations, but it does have a solution. Therefore, at least mathematically, there can be electic and magnetic field existing in a vacuum, even without any charge, current, or magnetic dipoles. The method for solving this differential equation is strange; I'm going to perform some random operations, but they do eventually lead towards a solution. The general stragtegy is to work with the third and fourth equations and try to decouple them, producing equations for \(E\) and \(B\) separately. In the process, the first two equations will be use to remove any divergence terms that arise while working with the other equations.

The first operation is to take the last of Maxwell's equation an apply the curl operator to both sides of the equation. On the right side, there is a curl and a time derivative. With reasonably assumption about \(B\text{,}\) I can interchange the order of these derivatives and then use Maxwell's third equation to replace \(B\text{.}\)

\begin{align*} \nabla \times (\nabla \times E) \amp = \nabla \times -\frac{\del B}{\del t} = \frac{-\del}{\del t} (\nabla \times B)\\ \amp = - \frac{\del}{\del t} \mu_0 \epsilon_0 \frac{\del E}{\del t}\\ \nabla \times (\nabla \times E) \amp = -\mu_0 \epsilon_0 \frac{\del^2 E}{\del t^2} \end{align*}

I've manage to get away from equations that involves \(E\) and \(B\) and produce a differential equation that only involves the electric field. This is progress. Now I want to adjust this equation into something I can actually solve. There is an identity for doing the curl twice: \(\nabla \times (\nabla \times F) = \nabla (\nabla \cdot F) - \nabla^2 F\text{,}\) where the Laplacian here means the Laplacian of each of the three components of \(E\text{.}\) (Feel free to verify this identity if you wish). IN this case, the first term is \(0\text{,}\) since \(\nabla \cdot E =0\text{,}\) so the remaining term is \(\nabla \times( \nabla \times E) = -\nabla^2 E\text{.}\) I'll replace the left side of the previous equation with this expression.

\begin{equation*} \nabla^2 E = \mu_0 \epsilon_0 \frac{\del^2 E}{\del t^2} \end{equation*}

A similar process exists starting with the third equation, applying the curl to both sides, replacing the \(E\) field with the fourth equation and simplifying the differential operators. It produces a very similar equation for \(B\)

\begin{equation*} \nabla^2 B = \mu_0 \epsilon_0 \frac{\del^2 B}{\del t^2} \end{equation*}

These are wave equations, so I expect that we will have wave solutions for \(E\) and \(B\text{.}\) The full solutions are very general, so I'm going to make some more assumptions which are justified by observation. First, I'll assume \(E \perp B\text{.}\) Second, I'll assume that \(z\) is the direction of propogation of the waves and the waves displacements are only in the \(xy\) plane. By readjusting coordinates under these assumption, I can write \(E = (E_1(z,t),0,0)\) and \(B = (0, B_2(z,t),0)\text{.}\) Then we have two differential equations in the scalar fields \(E_1\) and \(B_2\text{.}\)

\begin{align*} \frac{\del^2 E_1}{\del z^2} \amp = \mu_0 \epsilon_0 \frac{\del^2 E_1}{\del t^2}\\ \frac{\del^2 B_2}{\del z^2} \amp = \mu_0 \epsilon_0 \frac{\del^2 B_2}{\del t^2} \end{align*}

Now I've produced a wave equation in one spatial varaible and time. This has a well known solution using trig functions. I'll not repeat the steps of that solution here, but feel free to verity that the following functions do, in fact, satisfy the wave equations.

\begin{align*} E_1 \amp = a_1 \cos \left( z + \frac{1}{\sqrt{\mu_0 \epsilon_0}}t \right) + b_1 \sin \left( z + \frac{1}{\sqrt{\mu_0 \epsilon_0}}t \right)\\ B_2 \amp = a_2 \cos \left( z + \frac{1}{\sqrt{\mu_0 \epsilon_0}}t \right) + b_2 \sin \left( z + \frac{1}{\sqrt{\mu_0 \epsilon_0}}t \right) \end{align*}

What have I done here? I've shown that starting in a vacuum and making some geometric assumptions about the solutions, these trig function waves are valid electric and magnetic fields that can exist in a vacuum. These waves propogate in the \(z\) direction at a certain rate. The coefficient of \(t\) gives that rate. I can calculate the rough value of this coefficient (which depends only on the two fundamental constants).

\begin{equation*} \frac{1}{\sqrt{\mu_0 \epsilon_0}} = \frac{1}{\sqrt{8.86\times 10^{-12} \cdot 4\pi \times 10^{-9}}} = 2.997 \times 10^8 \frac{m}{s} \end{equation*}

The units also work out to recover a velocity. I'll put in the SI units for the permittivity and permeability constants.

\begin{equation*} \frac{1}{\sqrt{\frac{m \cdot kg}{s^2 \cdot A^2} \frac{s^4 \cdot A^2}{m^3 \cdot kg}}} = \frac{1}{\sqrt{\frac{s^2}{m^2}}} = \frac{m}{s} \end{equation*}

This is a familiar number: \(c\text{,}\) the speed of light. And light is exactly what this wave is: propogation of electromagnetic fields through a vacuum is light. At the time of Maxwell, light was not understood as electromagnetic radiation. Maxwell argued that his system predicted the electromagnetic nature of light, which was eventually proved correct. This is one of the major accomplishment of Maxwell's equations: just by solving them in a vacuum, they reproduce descriptions of light as an electromagnetic wave.

Lastly, this also explains by one of the three constants (as it was developed historically, this ended up being permeability) is not independent. Once you know the permittivity of space and the speed of light, the permeability can be calculated from those two constants. Again, this explains the strange \(4\pi\) in the definition of permeability: it is a derived constants, based on knowing the permittivity of space, the speed of light, and the SI unit conventions.