Deriving Ampere-Maxwell Law Equation | JL

[jeff1evesque]

Statement
I was wondering if anyone could help me derive the following equation,
$$\nabla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}$$

Background
Looking at wikipedia, the best I could come up with is that it is in differential form.

Question
Could someone help derive this. Or should I just except it as a definition, with a general explanation (if one could help with this instead)?

Thanks,JL

 

[Born2bwire]

The Maxwell's Equations are pretty much first principles for classical electrodynamics. They were not derived mathematically but rather via experimental results. Although you could technically derive one of the equations from the other three.

 

[jeff1evesque]

The Maxwell's Equations are pretty much first principles for classical electrodynamics. They were not derived mathematically but rather via experimental results. Although you could technically derive one of the equations from the other three.

Got it. Is there any chance you could give your interpretation (general of specific) of the equation I mentioned above.

Thanks,

Jeffrey

 

[Born2bwire]

Got it. Is there any chance you could give your interpretation (general of specific) of the equation I mentioned above.

Thanks,

Jeffrey

There isn't really anything that I would say that would be different from what wikipedia would say. It simply relates that the curl of the magnetic field, the solenoidal component of the magnetic field, is related the currents and time derivative of the electric flux density. In a source free environment, what you have is a relationship between a time-varying electric field and a magnetic field. Under the proper conditions, this, with the other equations of Maxwell, will allow you to find the wave equations that describe the propagation of electromagnetic waves. The differential form of Maxwell's equations are usually easier to work with mathematically but I feel that they hide the physical picture more. The integral forms usually have a more apparent physical meaning but the two forms are equivalent.

 

[jone]

The Maxwell's equations, are differential equations and no laws in the true sense. People just don't seem to get that!?

On the other hand, the Lorentz force law, the Coulomb's law, the Biot-Savart's law, the Faraday's law of induction as well as the conservation of charge are the fundamental laws of electromagnetics. These are experimental facts and can't be derived. From these you derive the Maxwell's equations in vacuum.

If you know the curl and and divergence of say the electric field E as well as the boundary conditions then you can also determine the E-field uniquely. This is the Helmholtz theorem. So we need one equation with the curl of B, and another independent one with divergence of B and similarly for E.

To derive Ampere-Maxwell's law we can start in magnetostatics. This is because we know that B is given by the Biot-Savart's law which one can start further develop to the general case (dynamic fields). For this we start by taking the curl of B, that is the curl of Biot-Savart's law. Applying some vector calculus to that will give you

$\nabla\times\mathbf{B} = \mu_o\mathbf{J}$

The above has to satisfy continuity equation (conservation of charge). Applying the continuity equation gives

$\nabla\times\mathbf{B} = \mu_o\mathbf{J} + \mu_0\epsilon_0\frac{\partial\mathbf{E}}{\partial t}$

That's the equation for vacuum. In matter you've maybe seen (and derived) the experssions for the polarization and the magnetization. Applying these give you the Ampere-Maxwell's equation in matter.

I know that many books in electromagnetics just post the Maxwell's equations in beginning claiming "this is how it is". I first learned electromagnetics from Griffiths' text, and he doesn't give you the Maxwell's equations until chapter 7. Instead he slowly derives the Maxwell's equations throughout the book. So I recommend Griffiths' book.

 

[turin]

$\nabla\times\mathbf{B} = \mu_o\mathbf{J}$

The above has to satisfy continuity equation (conservation of charge). Applying the continuity equation gives

$\nabla\times\mathbf{B} = \mu_o\mathbf{J} + \mu_0\epsilon_0\frac{\partial\mathbf{E}}{\partial t}$

That's the equation for vacuum.

I just want to correct a slight omission here, which may be confusing. Perhaps this is what jone means. The continuity equation alone is insufficient for this step, and Coulomb's Law is also required. This obviously applies in matter, because there is a J in the equation; however, it does not apply to time-variation - i.e. it only applies in the static case (in which, note that ∂_tρ=0). The extra term (added by Maxwell himself) "comes about" when you require the law to apply to time-varying fields and sources.