# Could someone explain the "revolving frame" to me, as it is used in basic NMR?

I am an undergrad intern at a national lab currently working with a basic proton NMR device. The device consists of two big coils which provide the static magnetic field, and a smaller coil, which sends both the "excitation" signal and receives the NMR signal. A week or so ago, my supervisor asked me to calculate the magnitude of the magnetic field of the inner coil.

Since I knew the maxiumum voltage of my signal, I chose to use Faraday's law of induction for a tightly wound coil of wire: $\mathcal{E}=-N\frac{d\phi_B}{dt}$. Knowing that the excited net magnetization vector $\vec{M}$ has a torque exerted on it (thanks to the static field), I reasoned that the magnetic flux through the smaller coil would be: $\phi_{B}=BA\cos\omega t$. Taking the derivative of this, I reasoned that the maximum voltage would be equivalent to the maximum of $-N\frac{d\phi_B}{dt}$:

$V_{max}=NBA\omega$.

I figured that from this equation; knowing the area, precession frequency, and number of coils; I would be able to find $B$ pretty easily.

But we measured B another way, using $\theta=\gamma B_1t_p$, and got a result that was half the size of the "Faraday way". At first, even my supervisor was confused, but then he quickly remembered that we forgot to remember that we were working in a "rotating frame".

For this reason, our supervisor said that the correct relation between max voltage and max flux was really $\frac{V_{max}}{2}=NBA\omega$.

Well, this baffled me. It still baffles me. He tried to explain to me that we can think of our signal as two arrows which oscillate in opposite directions in a circle, each arrow having a magnitude half the size of the actual thing. I know I'm using vague language right now, but that's because I don't get it. Why do I cut my max voltage in half. Even if we did use this rotating frame, wouldn't the max voltage be when the arrows are in phase, and add up together.

If you guys could help me out, that would be awesome. Let me know if you need any clarification on the setup of the experiment; I'll be happy to elaborate.

• By the way, NMR jocks usually call it 'the rotating frame', not 'the revolving frame.' Laser people often speak of the 'rotating wave approximation.' Commented Jun 1, 2014 at 23:45