ALTERNATING-CURRENT CIRCUITS — Edward Purcell & David Morin
Physics — E&M · dynamo-critical-frequency
Problem
Critical frequency of a dynamo *** A dynamo like the one in Exercise 7.47 has a certain critical speed ω0. If the disk revolves with an angular velocity less than ω0, nothing happens. Only when that speed is attained is the induced E large enough to make the current large enough to make the magnetic field large enough to induce an E of that magnitude. The critical speed can depend only on the size and shape of the conductors, the conductivity σ, and the constant μ0. Let d be some characteristic dimension expressing the size of the dynamo, such as the radius of the disk in our example.
((a)) Show by a dimensional argument that ω0 must be given by a relation of this form: ω0=K/μ0σd2, where K is some dimensionless numerical factor that depends only on the arrangement and relative size of the various parts of the dynamo.
((b)) Demonstrate this result again by using physical reasoning that relates the various quantities in the problem (R, E, E, I, B, etc.). You can ignore all numerical factors in your calculations and absorb them into the constant K.