W9_parallel_resonance.eps

210 Week 6: Moving Charges and Magnetic Force

or

vd=

E

B

(441)

We also know that:

I=nqvdA=nqA

E

B

(442)

Finally, we known that

V=Ew (443)

or

E=

V

w

(444)

so that

I=nqAV

Bw

(445)

We can then solve forn, the desired density of charge carriers:

n=

IBw

qAV

(446)

One can measureIandV directly.Bone can compute (although the Hall effect is actually often
used tomeasureB, as one can obviously turn this equation around and solve forBwith a strip
made from a material withknownn).wandAcan be directly measured with a ruler.

Best of all, we can finally see that the charge carriers in most metals areelectrons, that is, they
arenegative. Suppose that the carriers in the picture abovewereelectrons and negative. Then with
a current travelling to the right, they would actually be moving to theleft. The magnetic field would
then still divert themup, creating anegativestrip of charge on the upper edge of the strip and a
positive one on the lower. The electric field – for the same left-to-right current – would run from the
bottom to the top when the desired region of crossed fields established itself. This would make the
top of the strip at alowerpotential than the bottom, the opposite of what one gets with a positive
charge carrier.

Franklin’s Mistake is thus finally laid bare. Alas, the mobile charge in mostconductors is made
up of negatively charged electrons, the “cathode ray” particles discovered by Thomson. This is not
always the case, of course. Ionic fluid solutions (like salt water) canhave currents in whichboth
charge carriers are present. Also, in semiconductors the carriers can easily be quantum mechanical
“holes” in the electron density that have an effective positive sign.

As we can see, the magnetic force on discrete particles is a very useful thing! This by no means
exhausts the utility of magnetic fields for bending streams of charged particles around to make them
do our bidding.

In the last example, though, we went from a picture of single charges to one where we were
working with the coarse-grained continuum limit of a chargedcurrentonce again. Perhaps it is time
to think about the magnetic force on current carrying wires!

6.3: The Magnetic Force on Continuous Currents

If we contemplate our (by now) standard model for current in a uniform wire, where the currentI
is given by:

I=nqvdA=

∫

J~·nˆdA (447)