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GATE 2025 BT Q.64: solution and answer

GATE 2025 · BTQ.64 · 2 marks · NAT

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 × 10^8 T^-1 s^-1

(Round off to one decimal place)

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Answer and solution

ANSWER

B0 ≈ 23.5 T

Official answer: 23.2 to 23.8Source: Official answer key, GATE 2025 BT (IIT Roorkee)

  1. Recognize the governing relation: the NMR resonance (Larmor) condition is ω = γB₀, where ω is the angular precession frequency of the proton in the magnetic field B₀ and γ is the proton's gyromagnetic ratio.

  2. Relate angular frequency to the given resonance frequency ν: ω = 2πν, so ν = γB₀/(2π).

  3. Solve for the field strength: B₀ = 2πν/γ.

  4. Convert the given frequency to SI units: ν = 1 GHz = 1×10^9 Hz.

  5. Substitute the given values: γ = 2.675×10^8 T^-1 s^-1, so B₀ = 2π(1×10^9)/(2.675×10^8).

  6. Compute: 2π×10^9 = 6.283185307×10^9, and 6.283185307×10^9 / 2.675×10^8 ≈ 23.4885 T.

  7. Round to one decimal place as instructed: B₀ ≈ 23.5 T.

KEY IDEA

The NMR resonance condition (Larmor equation) relates the applied magnetic field to the resonance (Larmor) frequency through the gyromagnetic ratio of the nucleus: the angular frequency ω = γB₀, and ω = 2πν, so ν = γB₀/(2π), giving B₀ = 2πν/γ.

WATCH OUT FOR

Using ν directly in the Larmor equation ω = γB0 without converting ν to angular frequency (forgetting the factor of 2π), which gives a wrong field strength off by a factor of ~6.28.

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