Question

The gyromagnetic ratio of \(^{31}P\) nucleus is 17.23 MHz/Tesla. The Larmor frequency of \(^{31}P\) nucleus in an MRI scanner with a field strength of 3 Tesla is MHz. (Round off to one decimal place).

Hint:

Larmor frequency equals gyromagnetic ratio times field strength. Just multiply 17.23 by 3.

Answer 1

Correct Answer: 51.7

Step 1: Recall the Larmor equation.
Every nucleus with a magnetic moment (a "spin") precesses around an external magnetic field at a frequency proportional to the field strength. This is the Larmor frequency, given by
\[ f = \gamma \, B_0 \]
where \(\gamma\) is the gyromagnetic ratio of that nucleus (in MHz per Tesla, when expressed this way) and \(B_0\) is the strength of the main magnetic field, in Tesla.

Step 2: Identify the given values.
For \(^{31}P\), \(\gamma = 17.23\) MHz/Tesla. The scanner field strength is \(B_0 = 3\) Tesla.

Step 3: Multiply to get the Larmor frequency.
\[ f = 17.23 \times 3 = 51.69 \text{ MHz} \]
Rounded to one decimal place, this is 51.7 MHz.

Step 4: Sanity check.
For reference, hydrogen (\(^1H\)) has a gyromagnetic ratio of about 42.58 MHz/Tesla, so at 3 Tesla it resonates near 127.7 MHz. Phosphorus-31 has a smaller gyromagnetic ratio than hydrogen, so its Larmor frequency at the same field strength is smaller too, which matches our answer of 51.7 MHz being well below 127.7 MHz.

Final Answer:
The Larmor frequency of \(^{31}P\) at 3 Tesla is 51.7 MHz. \[ \boxed{f = 51.7 \text{ MHz}} \]