Using Tipler\u2019s notation starting on page 917, we begin with a particle of charge q and mass M, in a circular Bohr orbit of radius r.\u00a0 Never mind that there is no such thing as a circular electron atomic orbit; the results are valid.<\/p>\n
Particle velocity, v, and orbital period, T, are related by:<\/p>\n
vT = 2\u03c0r<\/p>\n
The charge traveling in a circle produces a current of:<\/p>\n
I = q\/T = qv\/2\u03c0r<\/p>\n
Multiplying this current by the enclosed area, the magnetic moment is obtained:<\/p>\n
m = (qv\/2\u03c0r) \u03c0r2<\/sup> = (qvr)\/2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0[1], 39-16, pg 918<\/p>\n Angular momentum for the mass is:<\/p>\n L = Mvr = M (2m\/q)<\/p>\n Rearranging and writing magnetic moment and angular momentum vectors in bold type:<\/p>\n m<\/strong> = (q\/2M) L\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>[1], 39-17, pg 918<\/p>\n Having used a circular orbit of a particle with charge and mass to obtain this formula, it can also be used when spin angular momentum is inserted for L<\/strong>. \u00a0For the electron, the spin angular momentum is known to be \u045b\/2.\u00a0 Also, still following Tipler, we can now insert e<\/em> for the charge of the electron in place of q, and me<\/sub> for the mass of the electron in place of the general mass M.\u00a0 There is just one problem when doing this however. \u00a0\u201cFor electron spin, the magnetic moment is twice that predicted by this equation.\u00a0 This extra factor of 2 is a quantum-mechanical result which has no analog in classical mechanics.\u201d ([1], pg 918)<\/p>\n The extra factor of 2 is the electron spin g factor, and is more precisely found to be:<\/p>\n ge<\/sub> = -2.002 319 304 3622\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0[2]<\/p>\n The magnetic moment of an electron can then be written as:<\/p>\n m = ge<\/sub> (e<\/em>\/2me<\/sub>) (\u045b\/2),<\/p>\n or equivalently:<\/p>\n \u03bcS<\/sub> = ge<\/sub>\u03bcB<\/sub>(S<\/strong>\/\u045b)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 [3]<\/p>\n This extra factor of 2.002 319 304 3622 shows that the gravitons which pass through electrons, and are not absorbed, serve as conjugate wave functions that double the expected magnetic moment of the electron.<\/p>\n \u00a0\u00a0<\/p>\n [1] Tipler, Paul A., Physics<\/span>, Worth Publishers, Inc., 1976<\/p>\n [2] http:\/\/physics.nist.gov\/cgi-bin\/cuu\/Value?gem|search_for=g+factor<\/a><\/p>\n