2021-USAPhO_plus-03
Great Hall
USAPhO · 2021 · Elektromagnetisme
Great Hall
The classical Hall effect was first measured by Edwin Hall in 1879, shortly after the publication of
Maxwell’s equations. In all parts of this problem, materials contain nV electrons per unit volume,
and each electron has charge qe < 0 and mass me . You may use these quantities in all of your
answers. We will begin by investigating the implications of the classical Hall effect.
1. An infinite plate in the xy plane, with thickness d in the z direction, is placed in a uniform
magnetic field B = B ẑ as shown. An electric field E = E is applied in the plane of the plate
and the system is allowed to reach a steady state.
y
x
B
E
(a) If the electrons have velocity v at steady state, what is the current density J? Recall that J
is defined as the total flow of charge through a unit cross-section area per unit time.
(b) In the Drude model, electrons are subject to both the Lorentz force and a damping force
v, where is a constant that depends on the material. In the above system, what is the
current density in the steady state? Give both the magnitude and direction of J, e.g. in polar
coordinates.
(c) Compute the electrical resistivity,
= lim
E
B0 |Jx |
and the transverse Hall resistivity
E
.
0 |Jy |
H = lim
(d) A Hall effect sensor detects the strength of magnetic fields. Consider the following circuit
consisting of a square plate of side length L and thickness d in a perpendicular uniform
magnetic field B.
B
VH
I
V
A
+ + + + + + + +
d
L
L
+
E
Copyright ©2021 American Association of Physics Teachers
2021 USAPhO+
longitudinal emf E is applied to the plate. At steady state, a Hall voltage VH is measured
across the plate due to the buildup of charge on either side of the plate. If the electrical
resistivity of the plate at zero magnetic field is , what is the Hall voltage VH and the
current I through the plate? Express your answer in terms of , E, B, and the dimensions
of the plate.
Experiments in the 20th century revealed that in many materials, the Hall resistivity could only
take certain discrete values. We will now show how this follows from Bohr quantization. (These
next parts are independent of the first part of the problem.)
2. A zero-resistance loop of wire of radius R and cross-sectional area Aw carries a counterclockwise
current I. A solenoid through the middle of the loops carries magnetic flux out of the page,
which we define to be the positive ẑ direction.
(a) If the electrons all have the same speed, what is the angular momentum of each electron?
(b) If we allow the flux in the solenoid to change, the usual, “mechanical” angular momentum L
of each electron is not conserved. Instead, a quantity called the canonical angular momentum,
Lcan = L + Cqe , for some constant C, is conserved. Find C.
(c) The Bohr quantization condition says that for a closed circular orbit, an integer number of
de Broglie wavelengths must fit in its circumference. The de Broglie wavelength is
=
h
pcan
,
where h is Planck’s constant, and pcan = Lcan /R is the canonical momentum. For a given
solenoid flux , what is the set of allowed mechanical angular momenta L?
(d) What is the minimum possible change in the magnetic flux for which the same set of mechanical angular momenta is allowed? This is known as the flux quantum.
3. Now, consider an annulus held perpendicular to a fixed, uniform external magnetic field B, and
suppose an additional, tunable magnetic flux threads the center of the annulus, with both
pointing out of the page. The annulus has a transverse Hall resistance RH (i.e., an EMF of E
around the annulus generates a perpendicular current E/RH via the Hall effect) and you may
neglect its self-inductance.
B
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2021 USAPhO+
10
(a) Suppose begins to increase slowly and steadily in time. After a short time, the electrons
will begin flowing steadily from one side of the annulus to the other. Do the electrons move
inward or outward? Justify your answer.
(b) If the threaded flux increases by , how many electrons pass from one edge of the annulus
to the other? You may use RH , among other variables, in your answer.
(c) As we showed in 2(d), if the magnetic flux changes by the flux quantum q , the allowed orbits
from Bohr quantization are unchanged. Quantum mechanics thus tells us that in conventional
materials, if the magnetic flux changes by q , an integer number k of electrons must pass
from one edge to another. What constraint does this place on the Hall resistance?
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