Linear and Quadrature $B_1$

• Imagine that you have a single fixed loop of wire in the lab, then
  

  $$\vec{B}_{1}^{lin} = b_{1}^{lin} \cos{\omega t} \; \hat{x} \\$$ (3.10)

  
  
  switching on/off at the right frequency, but what does this field look like in the rotating reference frame ???

• Relate rotating and lab basis vectors
  

  $$\hat{x}' = \hat{x}\,\cos{\omega t} - \hat{y}\,\sin{\omega t}= R_{z}(\omega t)\hat{x}$$

  $$\hat{y}' = \hat{x}\, \sin{\omega t} + \hat{y}\,\cos{\omega t} = R_{z}(\omega t)\hat{y}$$ (3.11)

  
  
  multiplying these equations by $\sin{\omega t}$ and $\cos{\omega t}$ allows us to figure out what $\hat{x}$ looks like in the rotating frame.

  
  
  
  
  
  
  
  

• end up with
  

  $$\hat{x} = \hat{x}'\cos{\omega t} +\hat{y}'\sin{\omega t}$$ (3.12)

  
  

• multiply 3.11 and 3.10 to get

  
  
  
  
  
  
  
  

  
  

  $$\vec{B}_{1}^{lin} = \frac{1}{2} b_{1}^{lin}[\hat{x}'(1 + \cos{2\omega t}) + \hat{y}'\sin{2\omega t}] \\$$ (3.13)

  
  

  
  
  
  
  
  
  
  

• Lose half of the magnitude of $\vec{B}_{1}$ in the rotating frame.

• Can add two linearly polarized coils in the lab frame to generate generate a $B_1$ that rotates so that it is constant in the rotating frame.
  

  $$\vec{B}_{1} = B_{1}\hat{x}'$$ (3.14)

  
  

• Usually work hard to use quadrature fields, since they are more efficient (IN CLASS QUESTION)