Hi Deewakar,

Analogy of B, H and M to electrostatics: You have got electric intensity E, electric induction D and polarization P. There is also relative permittivity eps_r and vacuum permittivity eps_0. In vacuum eps_r = 1, in any other medium, it is different than 1 (either higher or lower). This depends on how material reacts to presence of electric field (in vacuum it is = 1 because there is nothing that can react to field). Relations go like this:

D = eps_0*eps_r*E = eps*E

or

D = eps_0*E + P

I.e. E quantity refers to field that is imposed from "outside". D is quantity which takes into account reaction of material. Note that it is different quantity than E, even in vacuum! This is because Maxwell equations look better when this quantity is used instead. P represents virtually field generated by material as a response. In vacuum P = 0 (zero matrix).

Now in magnetism, B is like E; H is like D; and P_m is like P. However, note that their names are opposite than in electrostatics! B is induction of magnetic field (NOT intensity), H is intensity of magnetic field (NOT induction). P_M is magnetic polarization. It goes that P_m = mu_0*M, where M is magnetization you are interested in. mu_0 is permeability of vacuum (and mu_r is relative permeability of the magnetized medium). Relations between fields are the same as in the electrostatic case:

B = mu_0*mu_r*H = mu*H

or

B = mu_0*H + P_m = mu_0*H + mu_0*M

Note that in his case, it is the external quantity B (i.e. generated elsewhere like being the interstellar magnetic field, field generated by electromagnetic coils, etc. but NOT generated by the studied medium) which is on the left-hand side of equation. But from the viewpoint of names of quantities, it is the same (note induction/intensity reversal with respect to electrostatics). Historical reasons for that, but still used.

Quantity of mu_r represents how do the particles of medium (atoms, molecules, ions, electrons) behave in magnetic field. For example, plasma is diamagnetic - in magnetic field, charged particles rotate around field line in such a manner, as to mitigate the external field. These have mu_r < 1. Ferromagnetic media have Weiss domains, which are macroscopic structures of multiple atoms of same orientation of spins. In magnetic field, the domains align their resultant magnetic field with the external one and significantly amplify it as a result. These have mu_r > 10^4.

So to answer your question - you have your current carrying conductor. Use Biot-Savart law to calculate B in your point in space as generated by given conductor geometry (it is the best approach in your case, I have got experience with this of my own. If you are interested I can tell you how you can improve the calculation, or what other ways of B calculation exist). Now, if you have homogeneous isotropic gas (or whatever) inside your system, in first approximation your magnetic field B should be like:

B = mu_r*B_vac

where B_vac is what you get from Biot-Savart. Note that it is multiplied with relative permeability (which is close to 1 for most gases) and NOT the mu_0 = 4 pi*10^-7 !

I hope my explanation helped a little. With regards,

Tomas

Hi Deewakar,

I'm fully agree with Tomas explanation, but want to put your attention on the next points.

1. B, H and M are the vectors. In general case, magnetic permeability Mu is a tensor of 2nd rank. So along with equations B=Mu*H and B=H+M, vectors B and H could have different directions at the same point and pictures of “magnetic field’s lines” drawn for B and H could be significantly different.

2. B is magnetic field (other names: magnetic induction or magnetic flux density). It characterizes a force acting on unit charge moving with unit velocity (Lorentz low: F=q(E+[v x B]) inside of magnetic field B in the absence of electric field E=0. It also characterizes a force acting on dot-magnet with unit magnetic moment m: F=grad(m B). So B is experimentally measurable physical quantity.

3. H (names in use: magnetic field, magnetizing field, magnetic field intensity, magnetic field strength) could be applied as a characteristic of magnetic field, which arises at a definite point in free space (vacuum). It is in use for description of magnetic field, which is applied in an experiment by external magnetic field source or as a technical characteristic of a magnet itself. This concept could also sometimes simplify some theoretical calculations but they could be done without it too.

Regards,

Gregory.

The previous contributions explain already many details. Still I would like to add something. Making analogies between electric and magnetic fields is always dangerous. In standard textbooks it is usually agreed (as mentioned above) that B corresponds to E, but this is not the best choice for all situations. Furthermore the electric field is a vector, while B is pseudo vector. In detail there are many symmetry aspects to Maxwell’s equations which are more interesting for mathematicians than physicists. So lets drop this part here.

A more interesting point is that H is often considered an auxiliary quantity. This is due to the fact that usually H can be described as the gradient of a scalar potential. This would require the presence of magnetic charge. Up to now we assume that there is no magnetic charge in the sense of monopoles. This provided that the space we are talking about is single connected. However, elementary magnetic moments are well described by means of a circular current, which implies a vector potential. So let’s say B is “the real thing”. It has its origin either in (circulating) currents or elementary particles. The latter ones correspond to an infinite small magnetic moment. Having more than one, distributed over space, you get a dipole density, which you call magnetization. (Note: depending on the area of physics “magnetization” is used in different ways. In thermodynamics, e.g., it is related to dF/dH). So why use H. Well, as said above, it is mathematically more easy in many cases. In a single connected region, we can define an auxiliary scalar field, describing the same vector field as B. If in experiment you apply a field to a sample, it is easier to talk about H(external), as that is something you control. B on the other hand contains all the magnetic dipoles of your sample, which you do not control, well, not as much as you want. You might say that mu0 is just a conversion factor from B to H in vacuum, but things are actually more complicated. I do not go into detail here but look, e.g., for “impedance of vacuum”. Concerning mu, you may consider this as a tensor describing how local currents flow as a consequence of an external field. It is a (position dependent) tensor as it virtually transforms one vector (field) into another, i.e. field into current (density), which corresponds to magnetization. So as in the first reply, the mu (mu slightly bigger than 1) of your gas results in a local magnetization, therefore increasing your B.Tomas Marcovic already pointed out that this is only first order approximation.

In Electrodynamics there are three antisymmetric tensors of second rank B,D and J which have only three independent components. Thus these components can be associated to the three components of one ordinary vector and this mathematical object is named pseudovector because it doesn't transform as one real vector. There are also two fundamenta vectors H and E for the magnetic and electric fields. These basic physical quantities may be defined in vacuum (i.e. without matter) and related with matter trough the quantities M and P which are also two antisymmetric tensors and which can be associated to vectors only for homogeneous and isotropic materials having magnetic or electric dipoles. For M it is possible to have spins instead of classical magnetic dipoles for obtaining magnetic order because it is well known that the Bohr-van Leeuwen forbids to have a spontaneous magnetization M as it is needed for explaining ferromagnetic or antiferromagnetic order. Thus M and P are physically different because there is not a concept as spin for the electric dipoles and therefore for the polarization P, but playing a similar role in classical electrodynamics for taking into account the magnetic or electric dipolar order.

Finally we can see that the Lorentz force of electrodynamics use the electric vector field E and the pseudovector magnetic induction B. Therefore both quantities are associated in the same form as do the magnetic field H vector and the electric induction D pseudovector in the definitions of the electromagnetic fields in four dimensions. Two references for presenting this issue much deeper are:

1. J.D.Jackson, Classical Electrodynamics, John Wiley & Sons, third edition

2. D.Baldomir and P.Hammond, Geometry of Electromagnetic Systems, Clarendon Press, 1996.

Joe Foy (Arizona State) and I are finishing a new undergraduate E&M text this summer (Yale Press, F2015), and I have included a .pdf of couple of draft pages we are writing for an early chapter, where I drew a few illustrations. I agree with the answers given, they were very helpful for me as I have spent too much time sorting out the physical basis behind H lately.

The reason we use H at all is intertwined in the history of physics and the pole model of matter. If magnetic poles could move, like charges, then they would increase the pole density and the magnetic field would be irrotational, like the electric field. The only problem, of course, is that nature does not work this way.

In the 1800s, the pole model of magnetization was both popular and viable. It explained ferromagnetism better than any other model of the time. It was not until Heisenberg showed in 1928 that spin states and the exchange energy can correctly explain ferromagnetism that the pole model really died.

By then it was standard lore among physicists and engineers, and some ideas that relied on the pole model, such as the magnetic scalar potential, were already well-established.

As a related aside, in 1931, Dirac predicted the existence of a quantum of magnetic monopole. Dirac argued that if charge is quantized, so must be pole strength. Dirac also showed that the force holding two quanta of monopoles together must be about 4700 times stronger than that holding a proton and electron of the same distance, so it would be difficult to separate these monopoles if they actually exist. Except for one unreproducible result (Cabrera, B., 1982, Phys. Rev. Lett. 48, 1378–1381), no monopole has ever been found.

So, in short, avoid the use of H when you have a choice, except perhaps as the independent variable in the measurement of a hysteresis curve.

Dear Jonathan,

I am very sorry to say that you have many concepts that you need to rewrite . Let me to put you only two of them:

1. Ferromagnetism was explained by Weiss introducing a local field in 1907 and it is even today the Curie-Wess law what has its explanation and in fact as a generalization of the paramagnetism. It is true that in 1931, Heisenberg could explain what was the origin of such huge magnetic field just taking into account the interaction between spins,i.e. exchange coupling J positive ( negative gives the antiferromagnetic coupling). Nowadays we know much more couplings among them. But never can be explained with solenoids within the ferromagnetic material as you do.

a- P. Weiss, L'Hypothese du champ Moleculaire et de la Propriete Ferromagnetique, J. de Phys. 6, (1907) pp. 661-690.

b- W. Heisenberg, Z. Physik 69 (1931), pp. 287-297.

2. In Electromagnetism is necessary to have H and B or E and D (related by the constitutive equations) because they are different order tensors. For instance, it is wrong to write curl of B for the law of Ampere or curl of D in Faraday law. This calculation only would be allowed in vacuum and if the proper transformation of tensors is done. In fact H is the proper vector associated to the magnetic field and the histeresis cycle needs a new energy to be included: the anisotropy energy.

Look in my papers for the one: titled Geometrical information coded in Maxwell's equations: a review , where the electromagnetic tensorial representation is made in a simple form.

To Daniel Baldomir: "Look in my papers for the one: titled Geometrical information coded in Maxwell's equations: a review , where the electromagnetic tensorial representation is made in a simple form." - have you got a DOI?

The DOI is:

DOI 10.1108/03321641111101212

But it is better that you go to my papers because you can find more things related with your interests.

Since about 1960, we know that E and B have the same units, and it clarifies also H and M in classical electromagnetism. The International SI has allowed that in the CGS (while the MKS is, as yet, old).

For example, the book "Principles of Electrodynamics" by Melvin Schwartz, of Stanford Univ., of 1972, can serve to illustrate some of the points here. It is reviewed online.

It is easier to look at static B field in terms V=IR, where H is the voltage drop per unit length, current density is B, and 1/Mu(permeability) =resistivity so if path is L and A is area, V=H*L, I=J*A, R=resistivity *A*L.

Both "magnetic current"and EE's current need a close loop. So this analogy layouts nicely. So for a unit length, the B(current/A) =H(delta V)/(1/Mu)/A. Any B changes is because current thin out or condense based on curve equipotential lines in a uniform media.