W9_parallel_resonance.eps

Week 6: Moving Charges and Magnetic Force 209

Example 6.2.7: The Hall Effect

The final object of our study of the magnetic force on single charged particles is theHall effect,
the tendency of a current carrying wire in a magnetic field to build up avoltageacrossthe wire,
or conducting strip that is based on spontaneous charge separation in the conductor to create a
“region of crossed fields” where the electric field/force precisely balances the magnetic force (and
simultaneously creates a potential difference).

B

v

E

− − − − − − − −

+ + + + + + + +

d

Figure 71: In the Hall Effect, a magnetic field causes themobilecharge to accumulate on the upper
or lower edge of a conducting, current-carrying strip in a magneticfield. This in turn creates a
potential difference across the strip that can easily be measured.

The Hall Effect is a phenomenon that spontaneously occurs when a conductor carrying a current
is placed in a magnetic field that is perpendicular to the current. The effect is easiest to observe in
a ribbon shaped conductor that is relatively wide; one such is pictured above with widthw(top to
bottom) and cross-sectional areaA.

The Hall Effect can be used to make two very important classical measurements. First, as we
will easily see, we can finally determine thesign of the charge carriersin any given material, as
positive charge carriers (the particles that are physically moving tocreate the current) will actually
polarize the strip theopposite waythan negative ones. Second, it enables us to directly measuren,
the density of charge carriers in our basic model of conduction.

Here’s how it works. The strip is placed into a magnetic field perpendicular to the strip as shown
and a current is run through it. In the figure 71 above, we assumepositivecharge carriers as usual
so that the current is in thesamedirection as the drift velocity of the carriers, from left to right.

At first these moving charges experience a magnetic force that (right hand rule!) diverts them
into a curved trajectory to theleftas indicated by the dashed arrow on one of the charges. However,
charges near the top have nowhere to go andbuild upin a layer on the upper surface of the strip.
This charge layer creates an electric field that begins to oppose themotion of still more charge until
after a bit, the strip has equal and opposite amounts of positive (upper) and negative (lower) charge
on the top and bottom edges, the latter in the form of “holes” left from which the positive charge
carriers migrated.

The charges now move in aspontaneous region of crossed fields– the carriers in the middle move
in zero net force with the electric force down equal to the magneticforce up. This, in turn, creates
an electricalpotential differenceV across the strip that can be measured with a voltmeter, at the
same time that the current through the stripIis measured with an ammeter.

We know that for each charge, when this situation is established:

qvdB=qE (440)