Irodov – Problems in General Physics

Fig. 44.

is v' = vld'. Due to the Doppler effect the observed frequency is

, 111—(v/0 2 yid

The corresponding wavelength is X, = c/v = d (c/v — cos 0). When
0 = 45° and v c the wavelength is^ 0.6 p,m.
5.238. (a) Let vs be the projection of the velocity vector of the
radiating atom on the observation direction. The number of atoms
with projections falling within the interval vs, vs + dvx is

n (vx) dvx exp (—mv:12kT)•dvx.

The frequency of light emitted by the atoms moving with velocity

vs is o) = (.1) 0 (1 -I- vx/c). From the expression the frequency distri-

bution of atoms can be found: n (e) do) = n (vx) dvx. And finally

it should be taken into account that the spectral radiation intensity

n (co). (b) 6.o)/6) 0 = 2 V(2 In 2) kr/mc^2.

5.239. u— cin 1+ +17 V lcn' If V c, then

5.240. v = 1 /^2 c60 = 30 km/s.

5.242. 0' = 8°.

5.243. The field induced by a charged particle moving with

velocity V excites the atoms of the medium turning them into sources

of light waves. Let us consider two

arbitrary points A and B along the

path of the particle. The light waves

emitted from these points when the

particle passes them reach the point

P (Fig. 44) simultaneously and amplify

each other provided the time taken

by the light wave to propagate from

the point A to the point C is equal

to that taken by the particle to fly

over the distance AB. Hence, we

obtain cos 0 = v/V, where v = c/n is the phase velocity of light.

It is evident that the radiation is possible only if V > v, i.e. when

the velocity of the particle exceeds the phase velocity of light in

the medium.

5.244. Tmin = (nl V n 2 — 1 — 1)mc 2 ; 0.14 MeV and 0.26 GeV

respectively. For muons.

5.245. T — n cos0^ 1) mcz = 0.23 MeV.

I/ n 2 cos 2

5.247. T2 = bT il(b-FT j AX)= 1.75 kK.

5.248. Am = 3.4 Rm.

5.249. 5.10 9 kg/s, about 10 11 years.

5.250. T =V 3cRpluM =2.10^7 K, where R is the universal

gas constant, M is the molar mass of hydrogen.

V — (v/c) cos 0 — —WO cos 0

u (^) v )