In an NMR experiment, the RF source oscillates at 34 MHz and magnetic resonance of the hydrogen atoms in the sample being investigated occurs when the external field ${\overrightarrow{\mathbf{B}}}_{\mathbf{ext}}$has magnitude 0.78 T . Assume that ${\overrightarrow{\mathbf{B}}}_{\mathbf{int}}$and ${\overrightarrow{\mathbf{B}}}_{\mathbf{ext}}$ are in the same direction and take the proton magnetic moment component ${\overrightarrow{\mathbf{\mu }}}_{\mathbf{z}}$to be$\mathbf{1}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{26}}\mathbf{}\mathbf{J}\mathbf{/}\mathbf{T}$ . What is the magnitude of ${\overrightarrow{\mathbf{B}}}_{\mathbf{int}}$?

1. Oscillation frequency of the RF source, $\mathrm{f}=34\times {10}^6\mathrm{Hz}$
2. Magnitude of the external magnetic field, ${\mathrm{B}}_{\mathrm{ext}}=0.78\mathrm{T}$
3. The internal and external magnetic fields are in the same direction.
4. The component of the magnetic moment of proton,${\mathrm{\mu }}_{\mathrm{z}}=9.27\times {10}^{-24}\mathrm{J}/\mathrm{T}$

Magnetic resonance imaging (MRI) is a medical imaging procedure that creates detailed pictures of your body's organs and tissues by using a magnetic field and computer-generated radio waves.

Using the condition of total magnetic field and the derived equation of magnetic field from energy equations of Stern-Gerlach experiment and Planck's relation, the value of the internal magnetic field by substituting the given values.

Formulas:

The energy equation from the Stern-Gerlach experiment is,

$∆\mathrm{E}=2{\mathrm{\mu }}_{\mathrm{B}}\mathrm{B}$ ….. (1)

Here, the magnetic momentum of the proton is

The energy due to the Planck-Einstein relation is,

$∆\mathrm{E}=\mathrm{hf}$ ….. (2)

Here, the Plank’s constant is $\mathrm{h}=6.63\times {10}^{-34}\mathrm{J}.\mathrm{s}$.

At first from the given data, we can say that here the total magnetic field is given by:

$\mathrm{B}={\mathrm{B}}_{\mathrm{int}}+{\mathrm{B}}_{\mathrm{ext}}$ ….. (3)

Now, comparing equations (1) and (2), the value of the magnetic is as follows.

$$\begin{gathered} \mathrm{hf}=2{\mathrm{\mu }}_{\mathrm{B}}\mathrm{B} \\ \mathrm{B}=\frac{\mathrm{hf}}{2{\mathrm{\mu }}_{\mathrm{B}}} \end{gathered}$$

Now, by substituting the given data and the above equation of total magnetic field in equation (3), the value of the internal field is given by,

$$\begin{gathered} \mathrm{B}=\frac{\mathrm{hf}}{2\mathrm{\mu B}}-{\mathrm{B}}_{\mathrm{ext}} \\ =\frac{\left(6.63\times {10}^{-34}\mathrm{J}.\mathrm{s}\right)\left(34\times {10}^6\mathrm{Hz}\right)}{2\left(9.27\times {10}^{-24}\mathrm{J}/\mathrm{T}\right)}-0.78\mathrm{T} \\ =19\times {10}^{-3}\mathrm{T} \\ =19\mathrm{mT} \end{gathered}$$

Hence, the value of the magnetic field is 19 mT.