Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.
••• Semi-classical model of spin in a magnetic field. A large collection of proton spins in a magnetic field acts, in many ways, like a classical magnetic moment μ. Given that the torque on a moment in a field is given by Γ = μ × B, and that the classical angular momentum L is related to the moment μ by μ = γL, and finally that Γ = dL/dt, show that a moment μ in a field B will precess about the field direction with an angular frequency given by ω = γB.
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