Maxwell's profound achievement: the speed of electromagnetic waves

[Cleonis]

Lately I'm thinking about the question of how to convey how Maxwell was able to derive from first principles how fast electromagetic waves propagate. How is that possible at all?In Maxwell's time it was already well known, of course, that the electric field and the magnetic field are very much interconnected.
In retrospect we see that Maxwell contributed what amounts to a principle of equivalence. Let me call it Maxwell's principle .*

Maxwell hypothesized that changing of an electric field has the same physical effects as current has, and he suggested a thought experiment to arrive at a quantitive relation. Imagine a wire that is conducting a current, with at some point a capacitor in the circuit. As we know a magnetic field will surround that wire. With the capacitor we have that as the current proceeds an electric field builds up between the capacitor plates. That changing electric field will be surrounded by a magnetic field that is equivalent to the magnetic field that surrounds the current carrying wire.

In all, a changing electric field induces a magnetic field, a changing magnetic field induces an electric field. This mutual effect gives rise to a wave phenomenon. As all textbooks mention: Maxwell pointed out that the speed of this electromagnetic wave coincides with the speed of light.

http://www.phys.virginia.edu/People/Personal.asp?UID=mf1i"

Maxwell's equations imply the following relation:

[tex] c^2 = \frac{1}{\mu_0 \epsilon_0} [/tex]

Amazing. How come the electric constant and the magnetic constant relate to the speed of light?

As I understand it the following two factors are key:
- Maxwell's equations involve rate of change: rate of change of one field induces the other field.
- Electric field and magnetic field are in a certain ratio to each other. (Relativistic physics allows us to describe magnetism as a relativistic side effect of the Coulomb force. (See Daniel Schroeder's: http://physics.weber.edu/schroeder/mrr/MRRtalk.html" ) The relativistic nature is why the ratio of electric field and magnetic field has a factor 'c' in it.)

Let's say that at a signal emitter an electric field changes 1 volt in 1 nanosecond. Assume that change propagates away from that emitter at a velocity 'v'. That change is not an instantaneous jump, it's like an incline. The faster the propagation, the steeper the incline. When 'v' equals c the rate of change is just right, that is how Maxwell was able to conclude that the speed of electromagnetic waves coincides with the speed of light.* (I googled, there is already a 'Maxwell's principle', but it's used by just a few authors.)

 

[tiny-tim]

Hi Cleonis!

http://www.phys.virginia.edu/People/Personal.asp?UID=mf1i"

Maxwell's equations imply the following relation:

[tex] c^2 = \frac{1}{\mu_0 \epsilon_0} [/tex]

Amazing. How come the electric constant and the magnetic constant relate to the speed of light?

Yes, that is a good article, with some clear diagrams.

Fowler considers a plane pulse propagated at speed v perpendicular to an infinite sheet in the x,y plane (z = 0), by suddenly turning on a current of linear density I in the negative x direction at time t = 0 …

after time t, the effect is still zero for |z| > vt, but will be uniform (with E and B constant) for |z| < vt.

Fowler draws a rectangle Y of width y (he uses "L"), extending beyond |z| = vt, in the y,z plane (so a current yI goes through Y), and applies Maxwell's fourth equation, giving yI = -ε₀d/dt∫_Y E.dA = -ε₀d/dt(2vtyE) = -2ε₀vyE, or E = I/2ε₀v.

Then he draws a rectangle X of width x on both sides of z = vt (so two sides are perpendicular to E, and one side is still in the zero electric field) but not of z = 0, in the x,z plane, and applies Maxwell's third equation, giving xE = -d/dt∫_X B.dA = xvB, or E = vB.

(hmm … I've lost a minus somewhere )

We also know B = µ₀I/2, and so:

vB = vµ₀I/2 = I/2ε₀v,
or v² = 1/µ₀ε₀ ​

(but, until [STRIKE]Einstein[/STRIKE] Hertz, the connection with the speed of light, c, was only empirical: Maxwell simply noticed that numerically 1/µ₀ε₀ was extremely close to c )

 

[vanhees71]

But that was Maxwell's great breakthrough (of course, he has never come to the idea to invent something so clumsy as the SI of units, but that's another can of worms ;-))! Up to then it was not clear that light is an electromagnetic phenomenon. Optics was a topic not connected with electromagnetism, and Maxwell's discovery of the "displacement current" and thus the prediction of wavelike propagation of the electromagnetic fields with the speed of light lead to the hypothesis that light is nothing else than an electromagnetic wave (in a certain range of wavelenghts of about 400-800 nm). The direct experimental prove for the existence of such em. waves and the investigation of their properties, showing full accordance with Maxwell's predictions, is due to H. Hertz. The story of the classical electromagnetic theory has then been completed with Lorentz's theory of the electron and finally Einstein's (special) theory of relativity and Minkowski's electromagnetic model for macroscopic electromagnetics.

However, this is the historical point of view, which is of course very interesting and exciting. Another point of view is to look back to this history from the more modern point of view and looking at the theory from the now established point of view. Here, I'll restrict myself to the classical (i.a., non-quantum) side.

The argument starts with the fact that Einstein's theory of relativity as a space-time model contains a limiting speed, which cannot be exceeded for signal propagation. Within this space-time framework all theories have to comply with the underlying space-time structure. This immediately leads to local field-theoretical models or the interaction of particles (as opposed to action-at-a-distance models which are more natural in the older framework of Galilei-Newton space-time), and one of the most simple ones is the interaction of particles via vector fields. The most simple possibility then is a gauge invariant model, and this leads quite straight forwardly back to Maxwell's theory of the electromagnetic field. The speed of the waves from this point of view is the limit speed of Einstein-Minkowski space-time (or more generally Einstein-Hilbert space-time of General Relativity).

 

[Feldoh]

I've always found it interesting that Maxwell came about his "displacement current" by considering charge conservation.

I've also always found it interesting that we need monopoles for Maxwell's equations to be symmetrical.

 

[RedX]

, goes to show how little physics I know. Basically accept Maxwell's equations, then go through the derivation of the solutions on the very same day, without thinking about what the terms actually mean, or without using integral calculus.

But isn't that astonishing, that you can solve Maxwell's equations (in the case where sources are idealized - i.e., the fields created by the sources don't affect the sources)? The solution is right here:

http://en.wikipedia.org/wiki/Jefimenko's_equations#Electromagnetic_field_in_vacuum

Why can't one solve Newton's equations for an arbitrary force? Does this mean electrodynamics is easier than mechanics?

Anyways, something interesting in the Wikipedia link is this quotation:

There is a widespread interpretation of Maxwell's equations indicating that time variable electric and magnetic fields can cause each other. This is often used as part of an explanation of the formation of electromagnetic waves. However, Jefimenko's equations show otherwise.[3] Jefimenko says, "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents."[citation needed]

What do you think about this?

Anyways, I learned a lot from that website. That's pretty cool how Maxwell thought of the displacement current as dipole currents in aether induced by a gradually increasing electric field (negative charges flow one way, positive charges the other - and most importantly continue to flow as you amp up the field - so the total current is in the direction of the positive charge). Reminds me of hole-theory and Dirac - there is no sea of electrons of negative energy just as there is no aether, but it's brilliant and produces the correct results anyway.

Another similar paper is the introductory paper by Wilczek on fundamental constants:

http://arxiv.org/abs/0708.4361

that talks about the constants of electromagnetism.

 

[vanhees71]

Yes, Jefimenko's point of view is correct. Since Einstein and Minkowski we know that there are not separated electric and magnetic fields but only one electromagnetic field, which is uniquely determined by the Faraday-field-strength tensor [tex]F_{\mu \nu}[/tex], and the sources of this field is the electromagnetic current-four vector. That's why Maxwell's equations in free space should better be written as

[tex]\partial_{\mu} F^{\mu \nu}=j^{\nu}, \quad \partial_{\mu} {^*F}^{\mu \nu}=0[/tex].

Split in electric and magnetic fields with respect to an arbitrary inertial reference frame, these equations read

[tex]\vec{\nabla} \cdot \vec{E}=\frac{1}{c} j^0=\rho, \quad \vec{\nabla} \times \vec{B}-\frac{\partial \vec{E}}{c \partial t}=\frac{1}{c} \vec{j}[/tex]
[tex]\vec{\nabla} \cdot \vec{B}=0, \quad \vec{\nabla} \times \vec{E} + \frac{\partial \vec{B}}{c \partial t}=0[/tex].

I've used rationalized Gauß units (also known as Heaviside-Lorentz units), which in my opinion are best suited to the physical content of electromagnetism.

 

[RedX]

That's good. So is there no point to knowing Maxwell's equations? Once you have a solution to them, who needs the equations? In fact, once you have the solution, there should be an infinite number of equations that have your solution as a solution, so why Maxwell's equation in particular?

Anyways, great links. That was cool how Schroeder/Purcell derived the Larmor formula from just a picture with no calculus. But I think that model of the electromagnetic field as kinks in a picture is flawed pedagogically. If a charge is moving at constant velocity, how is it that at any instant, the electric field is radial coming out of the charge? At places really far away, the electric field should not know that the charge is moving at constant velocity, or otherwise this violates finite speed of information. The answer does seem to come from Jefimenko's equation however. The second term (see Wikipedia link) in the electric field looks like a correction to the first term (the retarded Coulomb term), and when added to the first term, gives an instantaneous Coulomb term rather than a retarded Coulomb term. How does this not violate the cosmic speed limit?

Anyways, having gone through the derivation of the Larmor formula, I tried to calculate how fast an electron falls into the nucleus. The acceleration is [tex]a=\frac{F_c}{m_e}=\frac{ke^2}{m_e r^2} [/tex], and the velocity² is [tex]v^2=r a [/tex], so setting:

[tex]P*t=\left(\frac{e^2 a^2}{6 \pi \epsilon_0 c^3}\right)t=\frac{m_e v^2}{2} [/tex]

I end up getting:

[tex]t=\left( \frac{4e^4k^2}{3r^3c^3 m_{e}^2} \right)^{-1}[/tex]

which comes out to 2x10^-20 seconds where I used the Bohr radius for the radius. Does this sound right?

 

[Born2bwire]

That's good. So is there no point to knowing Maxwell's equations? Once you have a solution to them, who needs the equations? In fact, once you have the solution, there should be an infinite number of equations that have your solution as a solution, so why Maxwell's equation in particular?

Jefimenko's Equations are great for explicitly showing the fields excited by the sources and the underlying physics but they are generally too cumbersome to work with. It first requires you to know all of the sources for a problem to find the fields which is not always easily known. The second problem is that it requires us to introduce sources into the picture and removes some of the advantages that going to the field picture gives us. Working with the fields allows us in many instances to solve a problem much easier. Try to think how you would go about using Jefimenko's Equation and the Lorentz Force to completely remove the field picture and derive Faraday's Law of Induction. It's easy to see conceptually how it comes about but to work explicitly from the source picture is very difficult.

Take for example a scattering problem. Let's be pendantic about it and look at the solution of a plane wave scattering off of a PEC sphere. First, we setup an infinite current sheet for our source. Use Jefimenko's Equation to find the excited fields along the surface of the PEC and then... Well... I don't know for sure how you would do this from an explicit source picture. You would have to find out the induced sources on the PEC sphere by the current sheet and then use Jefimenko's Equation to find the scattered field from these induced sources and add that to the incident field to get your total field.

From a Maxwell's Equations point of view we can use Maxwell's Equations to derive the boundary conditions of the fields on a PEC object. Then we can find the incident field along the surface of the PEC sphere from the initial conditions of the incident plane wave. Then we use the boundary conditions to find the scattered field that satisfies the boundary conditions (this is done by decomposing the plane wave into spherical waves and using mode matching to derive the scattered field as a summation of spherical waves). The point being here is that we do not need to explicitly introduce sources at all to find the total field. But Maxwell's Equations of course allow us to work with a source and field picture and so we do have the option of dipping into that part of the toolkit when needed (like we do with a moment method or finite element method).