Combine Angular Momentum and Magnetism to get an Equation of Motion

• If use $\mu = \gamma \vec{J}$ can re-write 2.6 as
  

  $$\frac {d \vec{\mu}} {dt} = \gamma \vec{\mu} \times \vec{B} \\$$ (2.9)

  
  
  This is a very important equation, because it explains precession which is necessary for us to be able to measure the MRI signal

• First step, draw a picture
  
  

• Now do a little figuring. Graphically, see that
  

  $$\vert d\vec{\mu}\vert = \mu \sin \theta \vert d \phi\vert$$ (2.10)

  
  
  (IN CLASS PROBLEM) and from the equation, can see that
  

  $$\vert d\vec{\mu}\vert = \gamma \vert\vec{\mu} \times \vec{B} \vert dt = \gamma {\mu}{B} \sin \theta dt$$ (2.11)

  
  

  
  
  
  
  
  
  
  

• Since, this problem is azimuthally symmetric, answer is going to be same no matter where you look as a function of $\phi$. Therefore, angular momentum is a constant.

• setting the previous two equations for $\vert d\vec{\mu}\vert$ equal illuminates the relation that we are looking for
  $$\begin{eqnarray*} \mu \sin \theta \vert d \phi\vert = \gamma {\mu}{B} \sin \theta dt \end{eqnarray*}$$
  
  
  
  
  
  Big result is that angular velocity of $\vec{\mu}$ is proportional to the magnetic field
  

  $$\omega \equiv \left\vert\frac {d \phi} {dt}\right\vert = \gamma B \\$$ (2.12)

  
  
  (IN CLASS QUESTION)

  
  
  
  
  
  
  
  

• Things to note
  • When spins are precessing, can measure this by Faraday induction.
  • Without external field there is nothing to precess about, and nothing to measure.
  • Even with exterman magnetic field, spin will find equilibrium aligned with $B_0$. Will need ways to perturb them.
  • Magnetic moment of a proton is tiny. Will need to rely on huge numbers to ever detect anything.