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39th International Physics Olympiad  Hanoi  Vietnam  2008
Theoretical Problem No. 2
CHERENKOV LIGHT AND RING IMAGING COUNTER
Light propagates in vacuum with the speed
. There is no particle which moves with
a speed higher than
. However, it is possible that in a transparent medium a particle
moves with a speed
higher than the speed of the light in the same medium
c
c
v
c
n
, where
is the refraction index of the medium. Experiment (Cherenkov, 1934) and theory
(Tamm and Frank, 1937) showed that a charged particle, moving with a speed
in a
transparent medium with refractive index
such that
n
v
n
c
n
>
v
, radiates light, called
Cherenkov light
, in directions forming
with
the
trajectory
an
angle
1
arccos
n
θ
β
=
(1)
A
B
where
c
=
v
.
1.
To establish this fact, consider a particle moving at constant velocity
c
n
>
v
on a
straight line. It passes A at time 0 and B at time
. As the problem is symmetric with
respect to rotations around AB, it is sufficient to consider light rays in a plane containing
AB.
1
t
At any point C between A and B, the particle emits a spherical light wave, which
propagates with velocity
c
n
. We define the wave front at a given time
as the envelope
of all these spheres at this time.
t
1.1. Determine the wave front at time
and draw its intersection with a plane
containing the trajectory of the particle.
1
t
1.2. Express the angle
ϕ
between this intersection and the trajectory of the particle
in terms of
and
n
.
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 Spring '11
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