Question

What is the ratio of Magnetic moment to Angular momentum?

Answer 1

How do I prove the relation between the vectors of magnetic moment μ⃗ μ→ and angular momentum L⃗ L→,

μ⃗ =γL⃗

μ→=γL→

?

Many text books and lecture notes about the principles of magnetism show the relation of μμ and LL as scalars only and then just state that the relation holds also for the vectors. An example: http://folk.ntnu.no/ioverbo/TFY4250/til12eng.pdf

μ=I⋅A=qtπr2=qv2πrmmπr2

μ=I⋅A=qtπr2=qv2πrmmπr2

(I=q/tI=q/t: current, AA: area of a loop, qq charge, t=2πr/vt=2πr/v: time of 1 rotation, vv: velocity of particle, mm: mass )

The angular momentum is L⃗ =r⃗ ×p⃗ L→=r→×p→ or L=mrvL=mrv and therefore

⇒μ=q2mL=γL.

⇒μ=q2mL=γL.

Why is this also true for the vectors? Is there a general explication by classical physics without the need of quantum theory?