Hyperfine splitting in \({ }^{1} \mathrm{H}\) is \(1420 \times 10^{6} \mathrm{~Hz}\) in the absence of a magnetic field where \(I . J . F . M_{F}\) are all good quantum numbers. In a moderately strong field I and \(\mathrm{J}\) are decoupled so that \(M_{I}\) and \(M_{J}\) are good quantum numbers.

(a) Estimate the field required to decouple \(\mathbf{I}\) and \(\mathbf{J}\) and by calculating the field that marks the magnetic energy equal to the hyperfine energy.

(b) In a magnetic field required that decouple I and \(\mathbf{J}\), how many levels are there?

Which of the levels, labeled \(\left(M_{I}, M_{J}ight)\), very linearly with \(B\) ?