In electron spin resonance, incoming electromagnetic radiation of the proper (resonant) frequency causes the electron’s magnetic moment to go from its lower-energy, or “relaxed,” orientation, aligned with the external field, to its higher-energy anti-aligned state. MRI is analogous. A quantity commonly discussed in MRI is the ratio of the frequency of the incoming radiation to the external magnetic field. Calculate this ratio for hydrogen. Note that the proton gyromagnetic ratio, ${\mathit{g}}_{\mathbf{p}}$, is $\mathbf{5}\mathbf{.}\mathbf{6}$ .

The gyromagnetic ratio of proton, $g_p=5.6$

The difference in energy between two spin orientations for a Hydrogen atom having one proton in its nucleus in the presence of a magnetic field B is,

$E=\frac{g_{p}e\hslash B}{2m_p}$ .....(I)

Here $\hslash$ is the reduced Planck's constant. $m_p$is the mass of proton having value, $m_p=1.67\times {10}^{-27}\text{kg}$ is the electron charge having value$e=1.6\times {10}^{-19}\text{C}$.

The energy of electromagnetic radiation of frequency f is

$E=2\pi f\hslash$ .....(II)

If the incoming radiation causes Hydrogen to go from lower magnetic moment to higher magnetic moment, the energy of the incoming radiation should be equal to the difference in energy of the two states. Thus equating equation (I) and (II) gives

$$\begin{gathered} 2\pi f\hslash =\frac{g_{p}e\hslash B}{2m_p} \\ \frac{f}{B}=\frac{g_{p}eB}{4\pi m_p} \end{gathered}$$

Substitute the values to get the following.

$\begin{array}{rcl}\frac{f}{B}& =& \frac{5.6\times 1.6\times {10}^{-19}\text{C}}{4\pi \times 1.67\times {10}^{-27}\text{kg}}\\ & =& 42.5\times {10}^6·\left(1\text{C×}\frac{1{\displaystyle \raisebox{1ex}{$kg$}\!\left/ \!\raisebox{-1ex}{$Ts$}\right.}}{1C}\right)\left(\frac{1}{1\text{kg}}\right)\\ & =& 42.5\times {10}^6\raisebox{1ex}{${\text{s}}^{\text{- 1}}$}\!\left/ \!\raisebox{-1ex}{$T$}\right.\\ & =& 42.5\raisebox{1ex}{$\text{MHz}$}\!\left/ \!\raisebox{-1ex}{$T$}\right.\end{array}$

Thus, the required ratio is $42.5\raisebox{1ex}{$\text{MHz}$}\!\left/ \!\raisebox{-1ex}{$T$}\right.$.