Question

Moment of inertia relation to magnetic moment formula

Answer 1

The easiest way to see the equality is to use a more general formula for the magnetic dipole moment of a particle. For a flat planar loop of current, it's true that μ=IAμ=IA, with the direction of the dipole normal to the loop. However, the more general case is that of a a volume current J⃗ J→ in some finite region of space. In this case, the the general formula for the magnetic dipole moment of the configuration is

μ⃗ =12∫r⃗ ×J⃗ d3r.μ→=12∫r→×J→d3r.

(Showing that this reduces to the above formula for a flat planar loop is left as an exercise to the reader.) If we further assume that the current density is due to a number of particles with number density nn, charge qq, velocity v⃗ v→, and mass mm, then we have current density J⃗ =nqv⃗ J→=nqv→; thus,

μ⃗ =12∫r⃗ ×(nqv⃗ )d3r=q2m∫r⃗ ×(nmv⃗ )d3r.μ→=12∫r→×(nqv→)d3r=q2m∫r→×(nmv→)d3r.

But nmv⃗ =ρv⃗ nmv→=ρv→, where ρρ is the mass density of the cloud; thus, the above integral can be rewritten as

μ⃗ =q2m∫ρr⃗ ×v⃗ d3r=q2mL⃗ .μ→=q2m∫ρr→×v→d3r=q2mL→.

QED.

Answer 2

The magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include: loops of electric current (such as electromagnets), permanent magnets, elementary particles (such as electrons), various molecules, and many astronomical objects (such as many planets, some moons, stars, etc).