W9_parallel_resonance.eps

Week 7: Sources of the Magnetic Field 233

q 1

1

q 2 v 2

v

r 12

F 12

21

Bin

B, F = 0

Figure 78: Newton’s Third Law fails for this arrangement because the field (and hence force) atq 2
due toq 1 iszerowhile the field (and hence force) atq 1 due toq 2 isnot zero!

This is, of course, offensive to all right minded individuals, who quite correctly view the failure
of energy or momentum conservation to be the failure of all of physics, a global inconsistency that
would cause us to observe all sorts of “magic” that we do not, in fact, observe in the world.

Historically, whenever momentum or energy haveappearedto be lost in a collision, we have (when
we looked carefully)foundthem again, often in an unexpected form. This case is no exception;in fact
it was probably the original case of the rule. Physics is quite safe, because momentumisconserved,
even though the momentum of a collection of massive particles interacting only electrically and
magnetically isnot! Where do you think the missing momentum and energy might be found?

7.3: The Biot-Savart Law

The magnetic field produced by a single charged particle has proven to be very interesting –too
interesting, in fact, for us to give a really complete treatment of it inan introductory course. As
noted above, the magnetic fields of individual particles were largely beyond the experimental reach
of eighteenth and nineteenth century physicists (which is where most of the electromagnetism we
learn in this book was discovered). In fact, the original magnetic fields studied were generated one
of two ways:

• They were “natural” fields generated by magneticobjects: Bar magnets, compass needles,
  magnetite mineral chunks, the Earth itself.


• They were the “artificial” fields generated bycurrent carrying wiresin the laboratory, under
  human-controlled conditions.

We will now focus on the second of these, current carrying wires, because historically Ampere
and Biot-Savart used measurements of forces between currentcarrying wires to establish most of
the important laws concerning the generation of magnetic fields, laws that in fact explain natural
magnetism as well.

Consider a current-carrying wire and the microscopic model for conduction that we worked with
in detail a few chapters back. In this model, a conductor can be viewed as an (electrically neutral)
collection of fixed charge and mobile charge. The mobile charge carriers all have chargeq(assumed
positive, although of courseqcould be either sign and in general will actually be negatively charged
electrons in metals). The density of carriers per unit volume isn. The wire has cross sectional area
A, and we will initially assume that the wire is “thin” compared to the distance to the point of
observation so we can avoid having to consider variations in the distance from a differential sized
chunk of the wire to the point of observation~r. The geometry is shown in figure 79.

We know from the last section that the field of justonecharge, say the one very close to the