Maxwell Equations: What are all the assumptions used in derivation?

[tommy555]

[SOLVED] Maxwell Equations: What are all the assumptions used in derivation?

I am trying to refute some of the theories of Tesla which are based on his idea that electromagnetic energy is also transmitted via a longitudinal wave. As far as I know Maxwell's equations do not support a longitudinal wave solution.

I would like to understand exactly what are all the assumptions that go into their derivation -- so that essentially for Tesla's theory to be correct one or more of those assumptions must be violated.

Thank you.

 

[heumpje]

Hi there,

To refute Tesla is very simple:
Suppose you have a plane wave traveling in the z-direction. In that case the solutions to the wave equations (derived from Maxwell's laws) have to be of the form:

vector(E)=vector(E₀)exp(i(kz-wt))

and

vector(B)=vector(B₀)exp(i(kz-wt))

That is we have a plane wave of monochromatic light with frequency w traveling in the z direction. vector(E/B₀) is the complex amplitude of the wave.
Plugging these solutions into div(E)=0 (there is no charge) and div(B)=0, we see that we have to have (E/B₀)_z=0 Since there is only a component in the x/y directions we have a transversal wave

 

[tommy555]

While this certainly makes sense, if I were trying to advocate Tesla then I would just assert that Maxwell's eqns are false under certain conditions. Can you help me to get behind what went into Maxwell's eqns to begin with?

Your insight would be greatly appreciated.

 

[Tyger]

I've looked at this a bit myself. If you had longitudinal EM waves it would force charged particles "off mass shell" i.e. electrons would change their rest mass under the influence of such waves. The fact that charged particles have to stay "on mass shell" determines the form of Maxwell's equations.

 

[heumpje]

Maxwell's equations are a set of laws that were already known. For instance div(E)=[rho]/e₀ is known as Gauss's law. Then there is Faraday's law and Ampere's law (the last one was adjusted by maxwell with an extra term). The only one without a name is div(B)=0 (an interesting one since it tells you there is no magnetic charge). What went into these equations is a lot of research and experimenting. As you might know Faraday was one of the best experimentors of his age. These laws can all be derived from elementary facts. You can look that up in any textbook on electrodynamics (you should look for electrostatics though).For instance David J. Griffiths "Introduction to electrodynamics" is a good book. It would take too much time to derive all that here (it takes Griffiths 285 pages to get to Maxwell's equations). Point is that you can derive these things from first principles (basic mathematics) and experimental fact.

 

[quartodeciman]

The following rules and guesses informed the Maxwell theory:

1. Coulomb rules - static electric charges and static magnetic poles have inverse square forces on static charges and static poles, respectively

2. Poisson rules - electrostatic and magnetostatic force fields can be derived from potential functions

3. Gauss rule - sum of signed electric charges in a volume is conserved

4. Biot-Savart rule - a small current element induces an inverse square magnetic field around it

5. Ampere rule - two small current elements have an inverse square force between them, dependent on their mutual orientation

6. Lenz rule - a flux of a magnetic field induces an opposing electromotive force

7. Faraday rule - a conductor moving across a magnetic field experiences an electromotive force

8. Maxwell guess - a flux of an electric field induces a magnetic field, without requiring actual current elements

.

Mathematically, Maxwell presumes that the fields can be represented by three-dimensional electric and magnetic field components, subject to three-dimensional vector calculus rules. Actual vector notation came later from Heaviside.

If longitudinal waves put in no appearance, it is because the Maxwell theory doesn't predict any.

 

[tommy555]

Thank you for all of the very interesting information.

Is it possible for Maxwell's equation to be traced back to only laws like conservation of energy, charge, etc?

 

[quartodeciman]

Obviously it is necessary to have some kind of interplay between electric and magnetic phenomena in order to produce the Maxwell equations. This can be direct or this can be subtle, but it must come in somehow.

Heinrich Hertz (who added charges and current vectors to the Maxwell equations) thought the result so fundamental and so total in consequences that he declared that these equations, along with Newton's laws, formed a complete foundation for classical physics, requiring no antecedent physical facts.

 

[Creator]

Originally posted by Tyger
I've looked at this a bit myself. If you had longitudinal EM waves it would force charged particles "off mass shell" ...

Tyger-
Why would longitudinal waves result in "off mass shell" charges ?Please explain.

Creator

P.S. Nice 8 point list, Quartodeciman.

 

[Creator]

Originally posted by tommy555


I would like to understand exactly what are all the assumptions that go into their derivation -- ...

Quarterdeciman gave a fairly good list.
It is interesting to realize that even though we quote his 4 eqns. in terms of the field vectors B and E, Maxwell himself derived many (if not most) in terms of what we call today the vector potential A, (which he referred to as electrodynamic momentum).

However, unbeknownst to many, there is one assumption that is usually omitted (or ignored) when considering Maxwell's derivation:

Maxwell derived his eqns. upon the firm conviction the ether is the MEDIUM in which electromagnetic phenomena takes place.

Creator

"The works of the Lord are great, studied by all who have pleasure therein"-- Inscribed in the archway of the entrance to James Clerk Maxwell's Cavendish Laboratory

 

[mr_whisk]

Hello, sorry too reopen an old thread - but I have some interesting information to share.

Is the Lorenz Condition (Gauge fixing) not an assumption?

Reading on the subject: http://www.teslaradio.com/pages/electrodynamics.pdf