W9_parallel_resonance.eps

Week 6: Moving Charges and Magnetic Force 211

(where, recall,nis the density of charge carriers,qis the charge per carrier,vdis the “drift velocity”

• the average velocity of the carriers in the wire,Ais the wire’s cross-sectional area) then we can
  add up the magnetic forces on all of the charges in a short (differential) length of wiredℓ:

dF~=nq(Adℓ)~vd×B~ (448)

We now do a clever thing. We’ll collect thenqvdAmagnitudestogether and makeI, and take the
directionof~vdand attach it todℓ, making it avectorpointing in the direction of the current in the
wire. The result is:
dF~=I(d~ℓ×B~) (449)

for a small (differential) segment of wire carrying a currentIin a magnetic fieldvB. Magnetic fields
exert forces on current carrying wires!

Toevaluatethe total force on any given current carrying wire is not, of course, likely to be
easyunless the wire has a very nice geometry, such as being astraight linein a uniform field or
acircular loop of currentin a uniform field. However, we can prove a very interesting result for
arbitrarycurrent loops that lets us understand how magnetic forces work on them to at least a
decent approximation, especially when those loops are “small” relative to everything else that is
going on. Let’s procede.

Example 6.3.1: The Magnetic Force and Torque on a Rectangular Current Loop (Magnetic Dipole)

a

b

each carrying I

N turns,

pivot axis

n

θ

θ

Fb

bin

bout

a

B

−F

b

θ

φ

Figure 72: The force and torque on ana×brectangular loop ofNturns, each carrying currentI,
in a uniform magnetic fieldB~areF~= 0 and~τ=~m×B~respectively.

In figure 72 you can see pictured a rectangular current loop withNturns, each carrying a current
I. When studying electrical currents and magnetic fields, using loopswith many turns is a cheap
and easy way to get a larger current than one’s power source can ordinarily support, as this is
effectively a current ofN Ion each leg of the circuit. If you’ve ever looked inside an electrical motor,
or transformer, or generator, or electronic device, you’ll almostcertainly see loops of reddish (epoxy
or enamel insulated) copper wire wrapped into loops with many turnsfor just this reason.