1.2. Express the angle between this intersection and the trajectory of the... · OlympiadBench #1329

Physics · optics

Problem

1.2. Express the angle

\varphi

between this intersection and the trajectory of the particle in terms of

n

and

\beta

. Given constants: CHERENKOV LIGHT AND RING IMAGING COUNTER Light propagates in vacuum with the speed

c

. There is no particle which moves with a speed higher than

c

. However, it is possible that in a transparent medium a particle moves with a speed

v

higher than the speed of the light in the same medium

\frac{c}{n}

, where

n

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

v

in a transparent medium with refractive index

n

such that

v>\frac{C}{n}

, radiates light, called Cherenkov light, in directions forming with the trajectory an angle

\theta=\arccos \frac{1}{\beta n} \tag{1}

where

\beta=\frac{v}{c}

. <image_1> 1. To establish this fact, consider a particle moving at constant velocity

v>\frac{C}{n}

on a straight line. It passes

\mathrm{A}

at time 0 and

\mathrm{B}

at time

t_{1}

. As the problem is symmetric with respect to rotations around

\mathrm{AB}

, it is sufficient to consider light rays in a plane containing

\mathrm{AB}

. At any point

\mathrm{C}

between

\mathrm{A}

and

\mathrm{B}

, the particle emits a spherical light wave, which propagates with velocity

\frac{c}{n}

. We define the wave front at a given time

t

as the envelope of all these spheres at this time. Context question: 1.1. Determine the wave front at time

t_{1}

and draw its intersection with a plane containing the trajectory of the particle. Context answer: <image_2>

Related theory

optics

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