Question about magnetostatic boundary condition.

[yungman]

My understanding of:

\int_S \nabla X \vec{H} \cdot d\vec{S} = \int_C \vec{H} \cdot d \vec{l} = I

Means the current I creates the magnetic field in the form of \nabla X \vec{H} instead of magnetic field creates the current I.

But in the boundary condition, it claims the tangential component of of \vec{H} is continuous if there is no surface current density \vec{J_S} on the boundary. This almost sounds like when an external magnetic field hit a conducting surface, current form on the surface result from the external magnetic field which do not agree with my assertion at the top that it is the current that cause the magnetic field.



I understand that varying magnetic field will cause electric flux in the Maxwell's equation:

\int_S \nabla X \vec{E} \cdot d \vec{S} = -\frac{\partial \vec{B}}{\partial t}

But this is in varying field condition not static condition as in my question.

Can anyone explain to me?

Thanks

Alan

[yungman]

Another point is that if it is true that a static magnetic field hit the perfect conductor surface and produce some current density. That means we can produce current in another circuit with just static magnetic field which we all know it is absolutely not true, you need a varying magnetic field to create electric field on the secondary circuit.

What is the current density in magneto static boundary condition means?

Thanks

Alan

[Born2bwire]

I don't think you can causally define the relationship without taking into account the actual problem. That is, if I have a current then this current can act as the source for the magnetic field. However, a magnetic field can also induce a current and act as the source too. It would just be better to note that in the general sense, the two are interconnected.

And thus, yes, an applied magnetic field does induce boundary currents at magnetically discontinous boundaries. These currents produce a secondary magnetic field that together allow for the total magnetic field to satisfy the appropriate boundary conditions. This is analogous to the fact that an applied electric field will induce bound charges along the surface of discontinuous permittivities.

One mechanism to produce a current using a static magnetic field is to physically change the magnetic flux through a closed coil of wire. This can be done by physically pulling the loop in and out of the magnetic field. In fact, this should be an obvious phenomenon when one considers how DC generators work. A DC generator uses static magnetic fields but the movement of the coiled armatures is what allows for the inducement of currents.

[yungman]

Thanks for the reply. I understand the idea of the interconnect of electric field and magnetic field in the time varying condition where

\nabla X \vec{H} = \vec{J} + \frac {\partial\vec{D}}{\partial t} \hbox { and } \nabla X \vec{E} = -\frac {\partial\vec{B}}{\partial t}

But I specified it is magneto static and all the books specified strongly that in non time varying condition, \vec{E} \hbox { and } \vec{H} and totally independent. I understand at t=0, the fields has to ramp up and there will be interlink between the two, but as it settle to the final value, the link will disappeared.

The more I think, the more I get confused.

I can understand on the electrostatic side that the surface charge on the conductor when the electric field hit the surface because the opposite chargo will form on the other part of the metal and the total charge in the metal piece remain neutral. But in the magnetic case, it is current density and by definition it is moving because

\vec{J}= \frac{\partial \rho}{\partial t}

I am confused!!! Please help.

Thanks

Alan

[Born2bwire]

I'm not sure what your source of confusion is. I made no statements about the situation being time-dependent. The statements above were made assuming magnetostatics. But your relationship between current and charge density is incorrect. First, the units for charge density should tip you off that it isn't right. The proper equation of charge conservation is

\nabla\cdot\mathbf{J} = - \frac{\partial \rho}{\partial t}

Regardless, this doesn't come into our situation. It simply falls out of the boundary conditions. Electric fields are sourced by charges, magnetic fields by currents. Due to the boundary conditions at an inhomogeneous interface, there are discontinuities in the electric field. This can be explained by the induction of a bound charge along the interface. Likewise, the magnetic field must also be discontinous and this can be explained by the induction of bound currents along the interface. These bounded sources create secondary fields that when added to the original field produce a total field that satisfies the boundary conditions. Classically, the inducement of bound currents is explained by the fact that materials have microscopic loop currents. These could be the orbiting electrons or something. The actual mechanism isn't entirely correct as we know from quantum mechanics but this is the assumptions taken in classical electrodynamics. Normally these loop currents are randomly oriented and thus there is no net field in the macroscopic regime. However, an applied magnetic field will induce alignment of these loop currents so that they orient along a direction with enough correlation to create a weak field. It is this weak field that is the secondary field and instead of representing the source for this field as a multitude of microscopic current loops we instead represent them (perfectly equivalently) as a single bounded surface current.

Again this is all analogous to what happens with an applied electric field in a dielectric. The applied field induces dipole moments and the equivalent effect is to give rise to bound charges on the surface.

[yungman]

My mistake on the \nabla \cdot \vec{J}.

I understand from the electric field causing surface charge and the opposite charge form on other part of the metal surface and the total charge is neutral if you take the whole piece of metal. One thing, my specific question is on metal surface and the charges are not bounded, I cannot agree with your assertion those are bounded charges ( dipoles).

Do you mean that the static magnetic field cause micro current loop to form? I just don't get the idea of a static magnetic will cause any current on the surface. A current density is current per unit area and current move!!!

Do you have any online article that can explain this?

Thanks

[Born2bwire]

Any textbook on electrodynamics like Griffiths will explain in detail about the polarization of dielectrics and the magnetization of magnetic materials.

The static magnetic fields do not cause these microscopic loop currents to form. They already exist as an inherent property of a material. The magnetic fields however cause the magnetic dipole moments of these loop currents to correlate along the direction of the applied field. This causes the dipole moments to create a non-negligible macroscopic magnetic field. Magnetic fields produce a force upon moving charges, currents. Thus their first order impact is upon any existing currents in a system.

Considering that a PEC is non-magnetic, the discussion of a perfect metal as a corallary to how electric/magnetic field polarize materials is moot. Anytime we are talking about magnetization we are doing so with respect to some imperfect metal or dielectric (or else we have to take on additional assumptions in say the case of superconductors).

I do not understand your problem with the fact that currents move either. Magnetostatics assumes static magnetic fields and static currents not static charges.

EDIT: Just to be clear, when we are talking about a perfect electrical conductor, it does not influence static magnetic fields. More specifically it is a bit of an ill-posed problem. The solution is that for a PEC, the static magnetic field inside the PEC must be constant. No additional conditions are derived and thus usually you just go with the fact that since the permeability is homogeneous the magnetic field (in the static case) would be continous as well. For superconductors, additional assumptions have to be taken so that it comes out that the magnetic field is zero inside the superconductor due to the Meissner effect.

[yungman]

Thanks

I forgot all about the obrital dipoles lining up due to magnetic field and create surface current.

Thanks