A dynamo like the one in Exercise 7.47 has a certain critical speed ?0. If the d

Question

A dynamo like the one in Exercise 7.47 has a certain critical speed ?0. If the disk revolves with an angular velocity less than ?0, noth- ing happens. Only when that speed is attained is the induced E large enough to make the current large enough to make the mag- netic field large enough to induce an E of that magnitude. The criti- cal speed can depend only on the size and shape of the conductors, the conductivity ?, and the constant ?0. Let d be some characteris- tic dimension expressing the size of the dynamo, such as the radius of the disk in our example.

(a) Show by a dimensional argument that ?0 must be given by a relation of this form: ?0 = K/?0? d2, where K is some dimen- sionless numerical factor that depends only on the arrange- ment 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.

Additional comments: for a dynamo of modest size made

wholly of copper, the critical speed ?0 would be practically unattain- able. It is ferromagnetism that makes possible the ordinary dc gen- erator by providing a magnetic field much stronger than the current in the coils, unaided, could produce. For an earth-sized dynamo, however, with d measured in hundreds of kilometers rather than meters, the critical speed is very much smaller. The earth’s mag- netic field is almost certainly produced by a nonferromagnetic

dynamo involving motions in the fluid metallic core. That fluid happens to be molten iron, but it is not even slightly ferromagnetic because it is too hot. (That will be explained in Chapter 11.) We don’t know how the conducting fluid moves, or what configuration of electric currents and magnetic fields its motion generates in the core. The magnetic field we observe at the earth’s surface is the external field of the dynamo in the core. The direction of the earth’s field a million years ago is preserved in the magnetization of rocks that solidified at that time. That magnetic record shows that the field has reversed its direction nearly 200 times in the last 100 million years. Although a reversal cannot have been instantaneous (see Exercise 7.46), it was a relatively sudden event on the geological time scale. The immense value of paleomagnetism as an indelible record of our planet’s history is well explained in Chapter 18 of Press and Siever (1978).

EXERCISE 7.47

In this question the term dynamo will be used for a generator that works in the following way. By some external agency – the shaft of a steam turbine, for instance – a conductor is driven through

a magnetic field, inducing an electromotive force in a circuit of which that conductor is part. The source of the magnetic field is the current that is caused to flow in that circuit by that electro- motive force. An electrical engineer would call it a self-excited dc generator. One of the simplest dynamos conceivable is sketched in Fig. 7.43. It has only two essential parts. One part is a solid metal disk and axle which can be driven in rotation. The other is a two- turn “coil” which is stationary but is connected by sliding contacts, or “brushes,” to the axle and to the rim of the revolving disk. One of the two devices pictured is, at least potentially, a dynamo. The other is not. Which is the dynamo?

Note that the answer to this question cannot depend on any convention about handedness or current directions. An intelligent extraterrestrial being inspecting the sketches could give the answer, provided only that it knows about arrows! What do you think deter- mines the direction of the current in such a dynamo? What will determine the magnitude of the current?

Explanation / Answer

Tangential speed = Angular velocity * Radius.

We know that to contiue to move without loosing the contact they need to have the same tangential velocities.

Vt = 2Pi * 1.5 = 3 Pi

Hence the angular velocuty fot the larger disk will be

w' = Vt/r = Pi rad/s

In this case Vt will become

Vt = r w = 1.5 Pi m/s

Hence the angular velocuty of the larger disk will be 1.5/3Pi = 0.5 rad/s

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