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Pan Pearl River Delta Physics Olympiad 2005

Jan. 29, 2005
Morning Session (9 am – 12 pm)

Q1 (5 points)
Two identical worms of length L are lying on a smooth and y
horizontal surface. The mass of the worms is evenly distributed
along their body length. The starting positions of the two
worms are shown in the figure. The coordinate of the center of A

worm-A is (0, 0). Worm-B then starts to climb slowly over
worm-A with their bodies always form an angle. After Worm- x
B has completely climbed over worm-A, what are the center B
positions of the two worms?

Q2 (13 points)
An air bubble of size 0.001m3 and a rigid tank of the same volume and mass as the bubble are
released at a depth of 2.0 km below the sea surface. Ignore friction. The temperature of the air
bubble remains the same at any depth. Air density at sea level is 1.21 kg/m3, and the
1    b 
b
dx
atmosphere pressure is 1.0 x 10 N/m . (hint: 
5 2
 ln  )
a
  x     a 
(a) What is the size of the bubble when it rises to the sea level? (3 points)
(b) Derive an expression for the net energy gained by the bubble and the tank at height h. (7
points)
(c) Find the final velocities of the bubble and the tank when they reach the sea level. (3 points)

Q3 (12 points)
A man with mass 0.5M is standing on a round table (disk
shaped, uniform thickness) rotating at angular speed  . The
mass of the table is 0.5M and the friction between the table and
the ground is negligible. The man carries with him 10 stones
each with mass 0.01M. The radius of the table is R and the man
is standing at a distance r (< R) from the center of the table. 


(a) Find the total angular momentum of the system. (4 points)

To slow down the rotation of the table, the man decides to

throw the stones outward from the table, each at speed v
(relative to him) and with angle  relative to the radial
direction (see figure).

(b) Determine the angular speed of the table after the man has thrown his first stone as a
function of angle  , and find the optimum angle  to slow down the table. (4 points)
(c) What is the angular speed of the table after the man has thrown all his stones, each time at
the optimum angle? (Leave your answer as the sum of multiple terms.) (4 points)

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Q4 (8 points)
A uniform rod of length L and mass M is resting in a smooth
hemisphere of radius R (>0.5L), as shown.
(a) Find the vibration frequency of the rod about its
equilibrium position. (4 points)
(b) In the vibration motion, the maximum deviation angle of the rod
from its equilibrium position is  max . Let the amplitude of the
contact force from the hemisphere to the rod at each end be N. The
difference between N when the rod is at  max and when the rod is
at its equilibrium position can be written as N  Mg max
2
. Find  .
(4 points)

Q5 (12 points)
 
The electric field of an electromagnetic (EM) wave is E  E0 x0 e i ( kZ t ) , where E 0 is a real
c
constant, and   ~ k . Here ω is real, c is the speed of light in vacuum, and n~ is the complex
n
dielectric constant of the medium.

(a) Briefly discuss what will happen to the EM wave amplitude as it propagates in the
medium if n~ is real, imaginary, or complex. (4 points)

(b) Find the magnetic field B , and the time-averaged (over one period) Poynting’s vector
 1  
S  ( E  B)  . (4 points)
0

dS 
(c) The quantity q  describes the loss of EM wave energy to the medium.
dz
Calculate q and briefly discuss the physical meanings of the results if n~ is real,
imaginary, or complex. (3 points)
(d) With reference to the results above, does an EM wave that decreases in amplitude
while propagating always loose energy to the medium? (1 points)

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Pan Pearl River Delta Physics Olympiad 2005

Jan. 29, 2005
Afternoon Session (2 pm – 5 pm)

Q6 (12 points)
Consider a uniform magnetic field B within the shaded region and pointing out of the paper
plane, as shown below.

B-field region B-field region

y
w

(a) (b) (c)

(a) Find the total force of the magnetic field on a closed thin wire coil carrying a steady
electric current I, all of which is inside the field region. The coil plane is within the
paper. (3 points)
(b) The total force of the magnetic field on the coil when part of it is outside the field
region can be expressed as F = wBI, where w is the distance between the two points
where the coil intersects the bottom edge of the field region, and its direction is either
upward or downward depending on the direction of the current. Find the value of . (3
points)
(c) A semicircle thin wire coil of radius r, resistance R, and mass m is falling down and out
of the field region. The plane of the coil remains in the paper plane, and its straight
edge remains parallel to the horizontal bottom edge of the field region. Ignore self-
inductance of the coil. Derive the differential equation for the distance between the
straight edge of the coil and the edge of the field region y (< R). If you have not found
 in (b) you may take it as a known constant in solving this part of the problem. (6
points)

Q7 (15 points) B
It is well known that when crossed electric and magnetic y 
VH j
fields are applied to a piece of semiconductor, a voltage VH
perpendicular to the direction of charge motion will be
induced (see figure). The phenomenon is called the Hall V
Effect. x

(a) Assume that the semiconductor is a square sheet of size W  W . The electric current is
due to the motion of positive charge carriers each carrying charge e. The surface
density of the carriers is n, and the conductivity of the semiconductor is . There is
also a negative charge background so there is charge neutrality everywhere except at
the side edges. The electric field is uniform in the semiconductor. A magnetic field B
is applied in the direction perpendicular to the sheet. When a voltage V is applied a
voltage VH across the two edges parallel to the current along the x-direction will be

induced, in addition to the electric current j . When the steady state is reached, find

the Hall Coefficient RH  VH / V . (Note that j is a unknown quantity) (6 points)

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Nowadays it is also known that for certain semiconductor structures, a spin-Hall effect will

also occur. The effect is associated with the magnetic moment m of the charge carriers. For
  
two dimensional structures it is known that an additional force FR   R (m  v ) (called

Rashba force) will act on the carriers, where v is the velocity of the carriers on the 2-

dimensional (X-Y) plane, and R is a constant. The magnetic moment m is restricted to point
 
perpendicular to the plane, i.e. m  mz . The external magnetic field is absence. Ignore the
magnetic dipole interactions between the carriers.

(b) Assume again that the electric field is uniform and y 

 z
its force along the x-axis is much stronger than FR , j
find the currents flowing in the y-direction in terms
of the voltage V and other parameters given in (a).
 x
V
How are the currents related to m ? (6 points)
(c) Due to collision with the boundary, the carriers with particular magnetic moment will
loose their sense of direction within a ‘life time’ after they reach the edge. In other
words, each second there are nm/ of carriers loosing their moment direction within a
unit length of edge, where nm is the surface density of the carriers still maintaining
 
their moment direction (  z ). Find the magnetization M near the edges. (3 points)

Q8 (23 points)
Electrorheological (ER) fluids, which are composed of small
dielectric spheres suspended in an insulating liquid, such as Plate
silicone oil, are materials that can transform from liquid-like
form to solid-like under an external electric field. A typical test ER Fluid
setting of ER fluids is shown in the figure, where ER fluid is
filled between two parallel conducting plates of area A separated
by a distance D. When no voltage is applied between the plates Plate
the ER fluid is liquid-like so the plates can moved horizontally
almost without friction.
When a voltage V is applied, the small spheres are polarized and aligned into vertical
columns, and to move a conducting plate relative to the other by a small displacement x
D f
requires a small force f. The shear modulus  is defined as   . The radius of the
A x
spheres is R (<< D), their dielectric constant is , and the volume fraction of spheres to fluid
is m. The dielectric constant of the liquid without the spheres is 1. Ignore gravity. You are to
find  in terms of the physical qualities given above.

(a) The first step is to find the polarization P of an isolated sphere 
 E0
in a uniform external electric field E0 . This can be done by
solving (a1) – (a3) below, and utilizing the known fact that
under such circumstance the polarization is uniform in the

sphere and parallel to E0 .

(a1) Find the electric field due to the polarization P at the center of the sphere. (3 points)
(a2) Find the total electric field inside the sphere. (3 points)

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 
(a3) The total induced electric dipole moment of a sphere can be expressed as p0  E0 .
Find the constant . (3 points)

(b) Treat each sphere as an ideal electric dipole located at the center of the sphere, and

assume that the dipole moment depends only on E0 . If you have not found  in (a3)
you may take it as a known constant in solving the following problems. (Hint: Keep
the expansion terms up to d2, where d is the length of the dipole.)

(b1) Find the electrostatic energies of two spheres in contact in the side-by-side and the
top-bottom configurations, as shown in the figures below. (4 points)
(b2) Find the electrostatic force of the conductor plate on the sphere that is in contact with
the plate. (3 points)
(b3) Find the restoring horizontal force between two spheres when the upper one in the
top-bottom configuration is displaced horizontally by a small distance a, as shown.
(3 points)


E0

(b1) (b2) (b3)

(c) Assume that under the applied electric field, all spheres form
continuous, straight, and single file thin columns between
the plates. According to your answers in (b1), do the
columns like to bunch together? Consider only the force
between adjacent spheres within a column, when the top
plate is displaced by a small distance x, the top sphere of
each column remains stick to the plate and is moved by the
same distance. As shown in the figure, each sphere in the
column below is then displaced uniformly relative to the one
just above. The bottom spheres remain fixed to the bottom
plate. Find the shear modulus . (4 points)

Pan Pearl River Delta Physics Olympiad 2005