Magnetic fields in matter Lec 5 stxaviers college ahmedabad.pptx

It largely talks about magnetic fiels in the matter

1. Magnetic fields in matter PH 5503 Unit III Lecture 5 Dr. Urvi Chhaya St. Xavier’s College Ahmedabad

2. Origin of bound currents Surface bound current : Physically, the bound currents on the surface of a magnetized material (and also in its volume for a non-uniform M) originate from the mis-cancellation of the microscopic currents inside the atoms which give rise to their magnetic moments. Surface bound currents are a result of the little current loops in a magnetized object failing to cancel at the boundary: the net effect is like a current flowing around the boundary of the object. we found that the field of a magnetized object is identical to the f ield that would be produced by a certain distribution of “bound” currents, J b and K b . Let us see how these bound currents arise physically. Figure depicts a thin slab of uniformly magnetized material, with the dipoles represented by tiny current loops. Notice that all the “internal” currents cancel: every time there is one going to the right, one is going to the left in the adjacent loop. However, at the edges there is no adjacent loop to do the canceling . The whole thing, then, is equivalent to a single ribbon of current I flowing around the boundary

3. Physically, the bound currents on the surface of a magnetized material (and also in its volume for a non-uniform M) originate from the mis-cancellation of the microscopic currents inside the atoms which give rise to their magnetic moments. As a model of how this works, consider a large L×L×L a uniformly magnetized material with M pointing in the z direction of the cube. For simplicity, let’s assume the material in question is a crystal with a simple cubic lattice — each tiny a × a × a cube occupied by a single atom, — and further more, assume each atom has the same magnetic moment m = a 3 M. The picture below shows a single slice of this cubic lattice along the (x, y) plane, or rather a small part of that slice: The red loops here stand for the current loops creating the atomic magnetic moments ‘m’. We do not know the radii or even the shapes of these loops or the currents which flow through them; we do not even know if the currents are line currents or volume currents. But for our purposes all such details do not matter, all we care is the net magnetic moment ‘ m’ of each atom.

4. As far as the macroscopic magnetic field B(r) is concerned, we may replace each atom with an a × a × a cube with the surface current flowing around the 4 vertical sides of the cube. Here is the picture of such a cubic model, or rather, of a small part of a single slice of the cubic lattice. What is this current, in terms of M ? Inside each atom, the current flows counterclockwise. But when we look at the boundary between two neighboring atoms, we immediately see that over that boundary, the currents of the two atoms flow in opposite directions. And since they have the same magnitude I = m/a 2 , they cancel each other! Thus, in the middle of the crystal all the atomic currents cancel each other and there is no net current. However, at the outer boundary of the crystal, there is no cancellation

5. Instead, there is un-canceled current I = m/a 2 flowing counterclockwise around the entire outer boundary of the crystal, or rather of single atomic layer of the crystal. If each current loop has an area a2 then the dipole moment due to a surface current I is equal to m = Ia 2 The volume of the current loop is a3 and therefore its dipole moment must be equal to m = Ma 3 This requires that the surface current of the current loop is equal to I = Ma (equating the rhs of above two equations) Since the magnetization is uniform, the current in each of the current loops will be constant and flowing in the same direction. Therefore, all volume currents cancel, and the only current remaining will be a surface current, flowing on the surface of the material.

6. The current flowing on the surface of the material will be equal to the current in each of the current loops. Therefore, the current density on the surface is equal to over the 4 vertical sides of the whole magnetized cube. In vector notations, the surface current on each side of the cube is where is the unit vector normal to the side in question. This expression also records the fact that there is no current on the top or bottom surface of the slab; here M is parallel to ˆn , so the cross product vanishes.)  

7. The above model explains the physical origin of the surface bound current. It does not have a volume bound current since we assumed a uniform magnetization inside the magnetic material. This bound surface current is a peculiar kind of current, in the sense that no single charge makes the whole trip—on the contrary, each charge moves only in a tiny little loop within a single atom. Nevertheless, the net effect is a macroscopic current flowing over the surface of the magnetized object. We call it a “bound” current because every charge is attached to a particular atom, and it produces a magnetic field in the same way any other current does. Thus bound currents are real and not just a mathematical construct

8. Volume bound currents occur when the magnetization is nonuniform. To model a non-uniform magnetization we should give different atoms of different magnetic moments m and hence different atomic currents. Consequently, Magnetization will vary from point to point and thus at the boundary of two neighboring atoms we would no longer have exact cancellation of their currents, and that would give rise to bound volume currents inside the bulk of the magnetized material. Suppose we look at the net current in the x direction, due to a difference in magnetization in the y and z directions:

9. When the magnetization is nonuniform, the internal currents no longer cancel. Fig. shows two adjacent chunks of magnetized material, with a larger arrow on the one to the right suggesting greater magnetization at that point. On the surface where they join there is a net current in the x -direction, At the interface between the two current loops, the net current in the x direction is: This corresponds to a current density contribution of:

10. By the same token, a nonuniform magnetization in the y -direction would contribute an amount - so In general, then in three dimension which is consistent with the result obtained earlier. Incidentally, like any other steady current, J obeys the conservation law: because the divergence of a curl is always zero.   Hence the origin of bound currents is understood. Both arise due to collective phenomenon

11. T he response of an object, be it any shape or symmetry, to magnetic fields is parameterized by Magnetization. Now we have understood that we can replace the volume magnetization by a volume bound current Jb and a surface bound current Kb and use them to calculate the potential and the field. In situations where there is some symmetry, we can use Ampère’s law to calculate the field from these bound currents, as this usually proves a lot easier than trying to do the integrals.