Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
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)
Official Answer Key:
23.2 to 23.8
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2025 Q.64)**
**1. Governing Concept & Definition**:
- **Topic**: Biophysical Methods & Nuclear Magnetic Resonance (NMR)
- The Larmor resonance frequency $\nu$ (in Hz) of a nucleus in an external magnetic field $B_0$ is:
$\nu = \frac{\gamma B_0}{2\pi} \implies B_0 = \frac{2\pi \nu}{\gamma}$
where $\gamma$ is the gyromagnetic ratio of the proton.
**2. Detailed Step-by-Step Calculation**:
- **Given**:
- Operating frequency $\nu = 1\text{ GHz} = 10^9\text{ Hz}$
- Gyromagnetic ratio of proton $\gamma = 2.675 \times 10^8\text{ T}^{-1}\text{s}^{-1}$
- Value of $\pi \approx 3.14159$
- **Calculation**:
$B_0 = \frac{2 \times \pi \times 10^9\text{ s}^{-1}}{2.675 \times 10^8\text{ T}^{-1}\text{s}^{-1}} = \frac{2 \times 3.14159 \times 10}{2.675} = \frac{62.8318}{2.675} \approx 23.488\text{ Tesla}$
**3. Final Answer**:
- **Calculated Value**: **23.5 T**
- **Official GATE Accepted Range**: **23.2 to 23.8**
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