Effective field in MRI when solving for Bloch equations I have a confusion regarding MRI signal formation. So, as I understand it, we need to solve the Bloch equations for the excitation and the relaxation stages:

So, in the excitation stage, let us assume that we have an RF field excitation which is applied along with the slice selection gradient. The bloch equations are given by:

$$
\frac{dM}{dt} = \gamma M \times B - R
$$

where $R$ is the vector accounting for the relaxation effects. Now, during excitation, the B1 field is on and as I understand it, the B vector is given as:

\begin{align}
B_x &amp;= B_1 \cos(\omega t)\\
B_y &amp;= B_1 \sin(\omega t)\\
B_z &amp;= z G_z 
\end{align}

The $B_z$ field is basically the gradient times the position offset from the origin and the $B_x$ and $B_y$ fields are the real and imaginary parts of the B1 waveform.

Now, when the B1 field is turned off, I think $B_x$ and $B_y$ components should be 0. However, let us assume that we have a readout gradient on when we are sampling the MR signal. 

What will contribute to the $B_z$ field now? Assuming that the RF excitation is with a 90 degree pulse and after the excitation the magnetization vector lies completely along the $Y$ axes.
In the rotating reference frame, $B_z$ from the static magnetic field $B_0$ is transformed away, hence $B_z = 0$ if no gradient field is applied. All gradients change the $B_z$ value, as you described it with the slice selection gradient. This gradient causes $B_z$ to increase when you move along $z$. Note that it is irrelevant if you do this from the center of the magnet or somewhere close to the bore - you will have the same $z$-dependence at each position $(x,y)$.

For the readout-gradient (assuming it is along left-right direction that we'll call $x$), the same is true: $B_z = G_Rx$. If you go from left to right, $B_z$ changes. In this case, it is independent of the position along the bore, so moving along $z$ does not have an influence on the value, if you stay at the same $x$ coordinate.

A gradient field always acts along the $z$-component of the magnetic field. This is always a bit tricky, since gradients can be "applied along each direction".