1. MAGNETIZATION AND SPIN MAGNETIC MOMENTS Among macroscopic objects we find those which have a permanent magnetic field, even if there are no obvious macroscopic currents. On the atomic and nuclear level we find particles, such as the electron, which themselves act as permanent magnets. These particles are said to have a magnetic dipole moment, or simply a magnetic moment for short. We would like to be able to express magnetic fields, B, in term of the vector potential, A. We will work out a specific case, but it turns out that it can easily be generalized. Consider the vector potential created by a circular current loop of radius a. There is a constant current I in the loop. The loop has a cross sectional area S so that I = JS, where J is the magnitude of the current density x y z dl r’  I P r  Calculate the field A at point p. The coordinates of P are (r,  The coordinates of the line element dl are (a, .

2. From equation 7 in the previous lecture we will write A as Note that J is tangential to the circle. We will assume that r >> a, so then the distance |r-r’| can be approximated by

3. m is called the magnetic dipole moment. Although we worked this out for a specific current distribution eqn (3) gives the correct leading order term for A for any current distribution. For macroscopic media then magnetic moment might be due to a domain. In the atom, the intrinsic spins of the electrons or other fermions will contribute to the vector potential A through eqn (3). We can define a magnetization M to be the magnetic dipole density.

4. The contribution to the vector potential dA is then for a distributed magnetization r’ r |r-r’| dA dm

5. Equation (5) can be recast using product rule 7 from reference 2. Since we can write The second integral in eqn (6) can be converted into a surface integral ( see ref. 2).

6. The conversion of the second volume integral to a surface integral is done like this ( again see ref. 2) The surface integral in eqn (7) becomes important if there is a discontinuity in the magnetization M. This happens in textbook examples. A physical magnetization will change continuously, so the first volume integral of eqn (7) is the only important term for our discussion of atomic nuclei. In the nucleus we have charged particles in motion, contributing to a conduction current J c (r’) that we included before, and now also a term coming from the magnetic moments of the fermions in the nucleus

7. The expression for A combining both these sources is We derived eqn (8) based on the assumption that the size of the dipole, a<<r, is small. For the elementary fermions, such as electrons and quarks, this should be valid because they are assumed to be point particles. The spins and magnetic moments of these fermions are understood in terms of relativistic quantum mechanics. Classical pictures can only be taken as a qualitative guide in developing a mental picture of spin or magnetic moments of the charged fermions. The current density which gives rise to the vector potential is thus,

8. REFERENCES 1)“Classical Electrodynamics”, 2nd Edition, John David Jackson, John Wiley and Sons, 1975 2) “Introduction to Electrodynamics”, 2 nd edition, David J. Griffiths, Prentice-Hall, 1989 ( This is an excellent text book. ) 3) “Electrodynamics”, Fulvio Melia, University of Chicago Press, 2001 4) “Relativistic Quantum Mechanics and Field Theory”, Franz Gross, John Wiley and Sons, 1993