Jackson Chapter 6junk

Jackson Chapter 6 Notes PhL 1.31.03 Chapter 6: xxx gaga 6.1 Faraday's Law. In this section, Jackson reviews the 1831 discovery of Faraday that if you change the magnetic flux surface integral on a current loop, a voltage is induced around that loop which causes a current to flow in that loop according to Ohm's Law. This voltage is usually written E and is historically called the "electromotive force". But it is the same kind of potential (voltage) we had in electrostatics, and it is as if a battery were in series with the loop, although the battery cannot be localized anywhere in the loop. We now know this voltage is due to the Maxwell equation xE = - /c. Jackson notes that the term Lenz's Law is used to describe the sign of the induced voltage -- it opposes the change in flux. In this little section, Jackson first assumes an unknown constant k in Faraday's law, as if xE = - k . He then considers a loop in motion and uses Galilean invariance to show that in the moving loop, charges are seeing a field E' = E + k(vxB). But from Biot-Savart (force on a current in a B field) we conclude that k = 1/c. Recall that Biot-Savart was developed in Chapter 5, as the force on a test current similar to Coulomb's law of the force on a test charge. So, the end result here is xE = - /c. Jackson does not claim to "derive" this equation. He just found what the constant must be to be consistent with Biot-Savart. 6.2 Energy in a B field. Here, Jackson goes through a long song and dance to arrive at 6.16, which says that the magnetic energy density is HB/8 . He does this in the following manner that I had much trouble following, it was geometric trouble, not vector trouble. First, consider a real wire loop as shown in the figure. Image this is a thin rectangular wire of finite but tiny cross-sectional area a (which he calls ). Now tile the interior of this loop with a fine mesh of tiny wire loops having the same cross sectional area. These loops are really squarish hollow puzzle pieces that fill the interior. Think of them as square hollow puzzle pieces that get slightly deformed as needed to fit the arbitrary initial loop. If you look at any one of these tiny loops, it has the full current I. The currents in these tiny loops cancel each other except at the boundary where we have the original loop. The big confusion is that one tends to think that the area is the area of the surface S shown in the figure, and then one starts thinking of J perpendicular to this area, etc. That is all completely wrong. Now focus on one of these tiny loops, and in particular, at a little d along a particular tiny loop. In this little chunk of the little loop, we know that J = I/a, and we know that J = (I/a) (d /d). Think of a*d as the volume of this little piece of wire which is part of this tiny loop, so write a*d = dV. Then we have shown that J dV = I d. This is the fact that I find to be unclear in Jackson's presentation. So if I were doing this presentation, I would start and I would retain I in all equations including the bottom one on page 174, and then in that one I would make the replacement I d = J dV. The understanding at that point with 6.12 would be that the volume integral was over the volume of the washer that forms the little loop, except that would be written (W) on the LHS in 6.12. Now we want to imagine that the mesh is so tight and small that the volume that is NOT in one of the tiny loops is negligible. So on the microscale, we get down to loops that are sort of FAT in appearance with a little tiny hole in the middle. A square donut with a square cross section having a very small square hole. We have so far computed the work done in this little loop due to a change in flux which is now being represented by a change in A, A. We next want to sum over all the loops of the mesh. In this sum the counterflowing boundaries all cancel, and the work we get then applies to the bounding wire. We then make the jump and say that the volume integral in 6.12 is over all space where we have current. I guess we could include the holes in this since no current in the holes, another way to do it. We finally end up with 6.15 assuming the field distribution is localized so you can drop the divergence integral. Then linear B,H relation does imply bottom of page 175 and we get the final result 6.16. It is maybe easiest to think of these tiny wires as having a rectangular cross section, so each tiny wire loop looks like a garden hose washer. Each tiny wire carries the same current I as the original wire, an , and loop at a small subloop. If the flux in that small loop changes B, a voltage is induced which, when multiplied by the current around the loop I = J Imagine 3D region of space containing some current distribution J. Now "tile" this entire 3D space with a set of square loop-objects each of which looks like this from the top: The loops have a thickness not visible in the picture, we might just set it equal to the width of the loops. If we look at one layer of the tiling from the top, this is what we see: We have allowed the loops to overlap as shown. Now, since this mesh is buried in a general current density J, we can say that on each edge of each loop we have J dV = I d, where dV = d * a, where a is the cross-sectional area of the tiling loop. The work that a changing B field does on one of these loops is given by (W) = (I/c) n B dA and we do what Jackson says at the bottom of page 174 to get this as a line integral. But then inside that line integral we replace I d with J dV so that now we get 6.12 where we have integrated . The edges of the donuts are 100% overlapping. Here is what one donut looks like from the top And here is what a mesh of 6 such donuts looks like from the top The donuts have some thickness not shown in these pictures, and in this manner we "fill" all of space in the area where we have some current density J.