– Dopple eltect lor < et
\[
\frac{\Delta e}{6}=\frac{v}{e} \cos \theta
\]

Fhere, is the welocity of a sources, 6 is the angle betwes the source’s motion dirsetion and the observation finm
– Doppler ettect is the genoral cass:
\[
\omega=\omega^{\frac{\sqrt{1-\beta}}{1-\beta \cos \theta}} \text {. }
\]
where $\beta=$ we.
– If $\theta=0$, the Deppler aftect is called radial, and if $\theta=a / 2$, transverse.
– Vavilow-Ciernkov eflect:
\[
\cos \theta-\frac{\epsilon}{n i n}
\]
where $\theta$ is the angle between the radiation propagation direction asd the valo eity vecter of of particle.
5.224. In the Fizean experiment on measurement of the velocity of light the distance between the gear wheel and the mirror is $l$ $=7.0 \mathrm{~km}$. the number of teeth is $z=720$. Two successive disappearances of light are observed at the following rotation velocities of the wheel: $n_{1}=283 \mathrm{rps}$ and $n_{2}=313 \mathrm{rps}$. Find the velocity of light.
5.225. A source of light moves with velocity $v$ relative to a receiver. Demonstrate that for $v<c$ the fractional variation of frequency of light is defined by $\mathrm{Eq}$. (5.6a).
5.226. One of the spectral lines emitted by excited $\mathrm{He}^{+}$ions has a wavelength $\lambda=410 \mathrm{~nm}$. Find the Doppler shift $\Delta \lambda$ of that line when observed at an angle $\theta=30^{\circ}$ to the beam of moving ions possessing kinetic energy $T=10 \mathrm{MeV}$.
5.227. When a spectral line of wavelength $\lambda=0.59 \mu \mathrm{m}$ is observed in the directicns to the opposite edges of the solar disc along its equater, there is a difference in wavelengths equal to $82=8.0 \mathrm{pm}$. Find the period of the Sun’s revolution about its own axis.
5.228. The Doppler eflect has made it possible to discover the double stars which are so distant that their resolution by means of a telescope is impossible. The spectral lines of such stars periodically become doublets indicatiag that the radiation does come from two stars revolving about their ceatre of mass. Assuming the masses of the two stars to be equal, find the distance between them and their masses if the maximum splitting of the spectral lines is equal to $(\Delta \lambda / \lambda)_{n}=1.2 \cdot 10^{-4}$ and occurs every $\tau=30$ days.
5.229. A plane electromagnetic wave of frequency $\omega_{0}$ falls normally on the surface of a mirror approaching with a relativistic velocity $V$. Making use of the Doppler formula, fiad the frequency of the reflected wave. Simplify the obtained expression for the case $V \ll C$.
5.230. A radar operates at a wavelength $\lambda=50.0 \mathrm{~cm}$. Find the velocity of an approaching aireraft if the beat frequency between the transmitted signal and the signal reflected from the aireraft is equal to $\Delta v=1.00 \mathrm{kHz}$ at the radar location.
5.231. Taking into account that the wave phase et $-k x$ is an invariant, i.e. it retains its value on transiticn from one inertial frame to another, determine how the frequency ond the wave number $k$ eatering the expression for the wave phase are transformed. Examise the unidimensional case.
5.232. How fast does a certain nebula recede if the hydrogen line $\lambda=43 \mathrm{~nm}$ in its spectrum is displaced by $130 \mathrm{~nm}$ toward lenger wavelengths?
5.233. How fast should a car move for the driver to perceive a red traffic light $(\lambda \approx 0.70 \mu \mathrm{m})$ as a green one $\left(\lambda^{\prime} \approx 0.55 \mu \mathrm{m}\right)$ ?
5.234. An observer moves with velocity $v_{1}=\frac{1}{2} e$ along a straight lise. In front of him a source of monochromatic light moves with velocity $v_{2}=\frac{3}{4} e$ in the same direction and aloag the same straight lise. The proper frequency of light is equal to $\omega_{4}$. Find the frequency of light registered by the observer.
5.235. One of the spectral lines of atomic hydrogen has the wavelength $\lambda=656.3 \mathrm{~mm}$. Find the Doppler shift $\Delta \lambda$ of that lise when observed at right angles to the beam of hydrogen atoms with kinetic energy $T=1.0 \mathrm{MeV}$ (the transverse Doppler effect).
5.236. A source emitting electromagnetic signals with proper frequency $\omega_{0}=3.0 \cdot 10^{10} \mathrm{~s}^{-1}$ moves at a constant velocity $v=$ $=0.80$ e along a straight line separated from a stationary observer $P$ by a distance $l$ (Fig. S.37). Find the frequency of the signals perceived by the observer at the zoment when
(a) the source is at the point $O$;
(b) the observer sees it at the point $O$.
5.237. A aarrow beam of electross passes immediately over the surface of a metallic mirror with a diffraction grating with period $d=2.0 \mathrm{~km}$ inseribed on it. The electrons move with velocity $v$, comparable to $c$, at right angles to the lines of the grating. The trajectory of the electroas can be seen in the form of a strip, whose colouring depends on the observation angle $\theta$ (Fig. 5.38). Taterpret this phenomenon. Find the wavelength of the radiation observed at an angle $\theta=45^{\circ}$.
5.238. A gas consists of atoms of mass $m$ being in thermodyaamic equilibrium at temperature $T$. Suppose $\omega_{0}$ is the aatural frequency of light emitted by the atoms.
(a) Demonstrate that the spectral distribution of the emitted light is defined by the formula
$\left(I_{6}\right.$ is the spectral intensity corresponding to the frequency $\omega_{6}$. $a=m c^{2} / 2 k T$.
(b) Find the relative width $\Delta \omega / \omega_{0}$ of a given spectral lise, i.e. the width of the line between the frequencies at which $I_{\sigma}=I_{y} / 2$.
5.239. A plase electromagnetic wave propagates in a medium moving with cosstant velocity $\boldsymbol{V}<\boldsymbol{c}$ relative to as iaertial frame $\boldsymbol{K}$. Find the velocity of that wave in the frame $K$ if the refractive index of the medium is equal to $n$ and the propagation direction of the wave coincides with that of the medium.
5.240. Aberration of light is the apparent displacement of stars attributable to the effect of the orbital motion of the Earth. The direction to a star in the ecliptic plane varies periodically, and the star performs apparent oscillations with is an angle $80-41^{\circ}$. Find the orbital velocity of the Earth.
5.241. Demonstrate that the angle $\theta$ between the propagation direction of light and the $x$ axis transforms on transition from the reference frame $K$ to $K^{\prime}$ according to the formula
\[
\cos \theta^{\prime}=\frac{\cos \theta-\beta}{1-\beta \cos \theta},
\]
where $\beta=V / c$ and $V$ is the velocity of the frame $K^{\prime}$ with respect to the frame $K$. The $x$ and $x^{\prime}$ axes of the refereace frames coincide.
5.242. Find the aperture angle of a cene in which all the stars located in the semi-sphere for an observer on the Earth will be visible if one moves relative to the Earth with relativistic velocity $\boldsymbol{V}$ differing by $1.0 \%$ from the velocity of light. Make use of the formula of the foregoing problem.
5.243. Find the conditions under which a charged particle moving uniformly through a medium with refractive index $n$ emits light (the Vavilov-Cherenkov effect). Find also the direction of that radiation.

Instruction. Consider the interference of oscillations induced by the particle at various moments of time.
5.244. Find the lowest values of the kinetic energy of an electron and a proton causing the emergeace of Cherenkov’s radiation in a medium with refractive iadex $n=1.60$. For what particles is this misimum value of kisetic energy equal to $T_{\text {mis }}=29.6 \mathrm{MeV}$ ?
5.245. Find the kinetic energy of electrons emitting light in a medium with refractive index $n-1.50$ at an angle $\theta=30^{\circ}$ to their propagation direction.