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Magnetic Multipole Moments Magnetic multipole moments We characterized the scalar potential V for a localized charge distribution at a point far from the charges with a multipole expansion of V: We can do the same with a vector potential due to a localized current distribution. The monopole moment turns out to vanish. The electric monopole moment was just the total charge of the distribution. Since we have no magnetic monopoles, we might expect the monopole moment to vanish. It is clear that the first term vanishes in the case of filamentary currents. That integral is just the vector displacement about a closed loop, which is zero. The same scenario will hold for a closed surface current. The proof that the monopole term vanishes for the case of a volume current density is given in Wangsness Section 19-1-1. It relies on a vector identity and use of the divergence theorem. The magnetic monopole moment always vanishes. The magnetic dipole term will be the dominant term in the multipole expansion. It is the only term we will consider. It will also turn out to be independent of the choice of origin of coordinates. This might also be expected since we found that the electric dipole moment was independent of the choice of origin when the total charge (electric monopole) vanished. The nice thing about the electric multipole expansion was that we could use it to characterize a charge distribution. So we would like to manipulate our magnetic dipole term to separate the source description from the position of the point at which we calculate the vector potential. We will find the following result, which defines the magnetic dipole moment. The proof of the relation given above for the case of a continuous volume current distribution is given in Section 19-1-2 of Wangsness. It is quite involved but straightforward so you can read it there. Here, we will go through the equivalent proof of the relation for the case of a filamentary circuit. We want to cleanly separate all dependence on the field point from the integration over the circuit. We can manipulate this form into an ordinary line integral form and apply Stoke’s theorem to accomplish this. Consider Stoke’s theorem in general first. So the desired separation is achieved here. So we can summarize the dominant dipole term in a common form for each type of current distribution. Given the vector potential, one can calculate the magnetic field associated with the idealized dipole contribution by taking the curl. In the electrostatic case, we had a simple picture of a physical dipole and a straightforward result for the leading dipole term of the electric field. The above would hold for a loop lying in the x-y plane at the origin. For a loop of radius a, small relative to r, Magnetism in the Presence of Matter Now we will go beyond the case of currents in vacuum. The concept of magnetic dipole will play the role that was taken by electric dipole when we discussed dielectric materials. Electric dipoles, as embodied by the charge separation of particles in atoms and molecules, felt the influence of external electric fields with the result characterized by the macroscopic polarization, and then described by bound volume and surface charge densities. With magnetic materials, it is ultimately the motion of charged particles (primarily electrons) that leads to the equivalent property of magnetization. Both orbital motions and intrinsic angular momentum (spin – quantum mechanical property) contribute to the magnetic properties of materials. The microscopic situation is complex and includes cases of paramagnetism (alignment of intrinsic magnetic moment by external field), diamagnetism (all materials, external field induces magnetic moment) and ferromagnetism (domains of aligned magnetic moments arising from electron spin interactions). We will describe magnetization in terms of volume and surface current densities. Take a small circular current loop as our prototype of a magnetic dipole. We can consider the influence on the loop (force and torque) of an external magnetic field. If the applied magnetic field is constant, the force on the loop vanishes. In the case of a position dependent magnetic field, the simplest way to derive an expression for the force is to approximate the value of the field at each point on the small loop via a Taylor expansion relative to the value of the field at the center of the loop (at the origin). The full details are given in Wangsness Section 19-4. But we will just quote the result because it is just like the situation we had in electrostatics: So this just serves to demonstrate that our magnetic dipoles can experience a force due to an applied field. Likewise, a magnetic dipole experiences a torque due to an applied magnetic field – even for a constant field. Here, consider that the applied field is constant. So we can see the possibility of alignment of magnetic dipoles within a material at the microscopic level. With polarized dielectrics, we used the polarization to find bound volume and surface charge densities. With magnetized material, we will find volume and surface current densities from the magnetization. The easiest way to extract those current densities is by consideration of the vector potential for the material. We know how to write the contribution to the vector potential of a loop of current due to its dipole moment. This is the dominant term for distances large relative to the loop size. We will leave the second volume integral intact but we will manipulate the first one into a surface integral. The volume over which we are integrating is the volume of the magnetized material. Use this expression to rewrite our result for the vector potential. This is the form we want. Recall we have seen the vector potential expressed in terms of current densities. This corresponds to precisely those forms, for equivalent surface and volume current densities. These surface and volume current densities, arising as a result of the magnetization of a material are referred to as equivalent, Amperian, magnetization, or bound current densities. We’ll use the term in Wangsness – magnetization current densities. Just as with the bound charge densities in dielectrics,these magnetization current densities are real currents. When the magnetization is uniform (curl M =0), there are only surface currents. By thinking of the aligned magnetic dipoles as small current loops, it is easy to see how the surface current density arises. For uniform magnetization, in the volume of the material, currents from adjacent “dipole loops” cancel. If the magnetization has a spatial dependence such that its curl does not vanish, then this cancellation is incomplete and the volume current density is nonvanishing. So we have a continuity statement again, but for the case of a current density associated with magnetization, we find that it cannot contribute to any charge accumulation. With dielectric materials, we progressed from the concept of polarized material as an assembly of dipoles described the the polarization, to a description in terms of equivalent volume and surface charge densities. With magnetic material, we go from the concept of magnetized material as an assembly of magnetic dipoles (think of as small current loops) described by the magnetization, to a description in terms of equivalent volume and surface current densities. The Biot-Savart law for the magnetic field is modified to include magnetization current densities: The subscript “outside” is a reminder that we developed this description of magnetized material starting with the multipole expansion of the vector potential. However, as long as we do not try to describe magnetic material at the microscopic level, the result also holds inside matter, just as we found for the case of dielectrics. Now we can do a bit of a summary by revisiting our results on magnetism for the vacuum case, with our new understanding of the addition of magnetic materials. We will extend our results for magnetism in this way. Magnetism so far, Introduction of H, and Some Boundary Conditions Adding magnetic materials, the magnetic field due to magnetization current densities arises due to the motion of charges, just as for free currents. So the form of the magnetic field is identical. Thus our result remains intact – after all, we interpret it as implying no static magnetic charges. So this result is modified when we include magnetic materials. It is important to note that this result was derived for the case of static fields. It will be further modified . We called the above result Ampere’s law in differential form. Like Gauss’ law in diff form, it has limits to its application. Both laws only work for situations where the fields are differentiable. So we saw that the electric field was discontinuous at an interface containing a surface charge. In that case, we had to use the integral form of Gauss’ law. Likewise, the diff form of Ampere’s law will not hold at the surface of a magnetized material where we have a surface current density. An obvious example is an infinite plane sheet of current, where the magnetic field is clearly discontinuous. Let’s be guided by what we did with the introduction of dielectrics in further manipulating Ampere’s law in diff and integral forms. We separated free and bound charge density. Just as we found a form of G’s law for displacement, we can find a form of Ampere’s law for H. Consider the flux of curl H through an open surface S. It is important to remember that these results were derived for steady currents – that is, using .
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