5.5: Magnetic Moments
- Page ID
- 1208
Consider a particle of electric charge \( v\) performing a circular orbit of radius \( r\) in the \( y\) plane. The charge is equivalent to a current loop of radius \( r\) in the \( y\) plane carrying current \( \mu\) of the loop is of magnitude \( z\) -axis. Thus, we can write
where \( {\bf x}\) and \( {\bf p} = {\bf v} /m\) , where \( m\) is its mass. We also know that \( {\bf L}\) is the orbital angular momentum. It follows that
Using the usual analogy between classical and quantum mechanics, we expect the above relation to also hold between the quantum mechanical operators, \( {\bf L}\) , which represent magnetic moment and orbital angular momentum, respectively. This is indeed found to the the case.
Spin angular momentum also gives rise to a contribution to the magnetic moment of a charged particle. In fact, relativistic quantum mechanics predicts that a charged particle possessing spin must also possess a corresponding magnetic moment (this was first demonstrated by Dirac--see Chapter 11). We can write
where \(g\) is called the gyromagnetic ratio. For an electron this ratio is found to be
for an electron (here, \(e> 0\)).
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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