Q.64
An NMR spectrometer operating at proton resonance frequency (ν) of
1 GHz will have a magnetic field strength of __________ Tesla (T).
The gyromagnetic ratio for proton,
γ = 2.675 × 108 T−1 s−1.-
(Round off to one decimal place)
The Larmor equation relates proton resonance frequency to magnetic field strength in
NMR spectrometers. For a 1 GHz proton resonance frequency and
gyromagnetic ratio γ = 2.675 × 108 T−1 s−1,
the required magnetic field strength is 23.5 T.
Larmor Equation Basics
The Larmor equation is given by:
ν = (γ B0) / (2π)
where:
- ν = resonance frequency (Hz)
- B0 = magnetic field strength (T)
- γ = gyromagnetic ratio (T−1 s−1)
Rearranging the equation:
B0 = (2πν) / γ
Step-by-Step Calculation
Step 1: Convert Frequency to Hertz
1 GHz = 1 × 109 Hz
Step 2: Calculate 2πν
2πν = 2 × 3.1416 × 1 × 109
= 6.2832 × 109 rad s−1
Step 3: Divide by Gyromagnetic Ratio
B0 = (6.2832 × 109) / (2.675 × 108)
= 23.49 T
Step 4: Final Answer
Rounding to one decimal place:
✔ B0 ≈ 23.5 Tesla
Interpretation
This value matches modern ultra-high-field NMR spectrometers, where a
1 GHz proton resonance corresponds to approximately 23.5 T
magnetic field strength.
Common Misconceptions and Option Analysis
Typical exam errors arise from incorrect unit handling or misuse of the Larmor equation.
| Common Error | Calculation | Result (T) | Why Incorrect |
|---|---|---|---|
| Omitting 2π | ν / γ | ≈ 3.7 | Ignores angular frequency conversion |
| Using γ / 2π incorrectly | 109 / (γ / 2π) | ≈ 3.7 | Misapplies gyromagnetic ratio definition |
| Frequency in MHz | 1 × 106 Hz | 0.023 | Incorrect GHz to Hz conversion |
| Correct Method | 2πν / γ | 23.5 | Physically correct for proton NMR |
Key Takeaway
Always check whether the gyromagnetic ratio is defined for angular frequency
(ω = γB). Including the 2π factor is essential for accurate
magnetic field calculations in NMR spectroscopy.
