(Solved): A typical liquid NMR sample is contained in a glass tube with diameter \( d=5 \mathrm{~mm} \) and ...

A typical liquid NMR sample is contained in a glass tube with diameter \( d=5 \mathrm{~mm} \) and length \( l \simeq 2 \mathrm{~cm} \). The first sample we will look at is a mixture of \( 5 \% \) water and \( 95 \% \) heavy water by volume. \( { }^{4} \) Calculate the following for such a sample: In a magnetic field \( B_{0}=1.5 T \), we want to know the equilibrium (thermal equilibrium) polarization of our deuterated water sample from the previous problem. Assume the populations are Boltzmann distributed and at room temperature. Compute the following values in units given : (a) The number of Hydrogen nuclei (protons) in the spin-up and spin-down states. (b) The net magnetic moment in units \( [\mathrm{J} / \mathrm{T}] \). (c) The order of magnitude of the magnetic field strength at a distance of \( 2 \mathrm{~cm} \). Hint: use the approximate formula \( |B| \simeq \frac{\mu_{0}}{4 \pi}|m| / d^{3} \) in SI units.