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5.236. A source emitting electromagnetic signals with proper frequency (00^ 3.0.1010 s-1moves at a constant velocity v = 0.80 c along a straight line separated from a stationary observer P by a distance (^1) (Fig. 5.37). Find the frequency of the signals perceived by the observer at the moment when (a) the source is at the point 0; (b) the observer sees it at the point 0.
op
Fig. 5.37. Fig. 5.38.
5.237. A narrow beam of electrons passes immediately over the surface of a metallic mirror with a diffraction grating with period d = 2.0 tim inscribed on it. The electrons move with velocity v, comparable to c, at right angles to the lines of the grating. The trajectory of the electrons can be seen in the form of a strip, whose colouring depends on the observation angle 0 (Fig. 5.38). Interpret this phenomenon. Find the wavelength of the radiation observed at an angle 0 = 45°. 5.238. A gas consists of atoms of mass m^ being in thermodynamic equilibrium at temperature T. Suppose coois the natural frequency of light emitted by the atoms. (a) Demonstrate that the spectral distribution of the emitted light is defined by the formula I.= Ioe-a(1--(0100)2, (I, is the spectral intensity corresponding to the frequency coo, a = mc2/2kT). (b) Find the relative width O w/w0of a given spectral line, i.e. the width of the line between the frequencies at which I. = / 0/2. 5.239. A plane electromagnetic wave propagates in a medium moving with constant velocity V < c relative to an inertial frame^ 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 within an angle 60 = 41". Find the orbital velocity of the Earth.
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(5.7d) u hco3^1 ,11e3 (^) efie)/kT (^) •
5.241. Demonstrate that the angle 0 between the propagation direction of light and the x axis transforms on transition from the reference frame K to K' according to the formula
cos 0' =
cos 0— p 1-13 cos 0 '
where p = V/c and V is the velocity of the frame K' with respect to the frame K. The x and x' axes of the reference frames coincide. 5.242. Find the aperture angle of a cone 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 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 emergence of Cherenkov's radiation in a medium with refractive index n = 1.60. For what particles is this minimum value of kinetic energy equal to Tmin= 29.6 MeV? 5.245. Find the kinetic energy of electrons emitting light in a medium with refractive index n = 1.50 at an angle 0 = 30° to their propagation direction.
5.7. THERMAL RADIATION. QUANTUM NATURE OF LIGHT
Me= + u, (5.7a)
where u is the space density of thermal radiation energy.
- Wien's formula and Wien's displacement law: u. = ca3F (co/T), T? m = b, (5.7b) where Xmis the wavelength corresponding to the maximum of the function uk.
- Stefan-Boltzmann law: Me = crT4, (^) (5.7c) where a is the Stefan-Boltzmann constant.
- Planck's formula:
- Einstein's photoelectric equation: 40= (^) A+ mu^2 rx. (5.7e)
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0. 43 as
5.253. A cavity of volume V = 1.0 1 is filled with thermal radia- tion at a temperature T =^ 1000 K. Find: (a) the heat capacity Cv; (b) the entropy S of that radiation. 5.254. Assuming the spectral distribution of thermal radiation energy to obey Wien's formula u (o), T) = A w3 exp (—acolT), where a = 7.64 ps•K , find for a temperature T = 2000 K the most probable (a) radiation frequency; (b) radiation wavelength. 5.255. Using Planck's formula, derive the approximate expressions for the space spectral density uo, of radiation (a) in the range where ho) <^ kT^ (Rayleigh-Jeans formula); (b) in the range where No >> kT (Wien's formula). 5.256. Transform Planck's formula for space spectral density u. of radiation from the variable a) to the variables v (linear frequency) and X (wavelength). 5.257. Using Planck's formula, find the power radiated by a unit area of a black body within a narrow wavelength interval AX = = 1.0 nm close to the maximum of spectral radiation density at a temperature T = 3000 K of the body. 5.258. Fig. 5.40 shows the plot of the function y (x) representing a fraction of the total power of thermal radiation falling within
Fig. 5.40.
the spectral interval from 0 to x.^ Here^ x =^ X/X„, (X,„ is the wavelength corresponding to the maximum of spectral radiation density). Using this plot, find: (a) the wavelength which divides the radiation spectrum into two equal (in terms of energy) parts at the temperature 3700 K; (b) the fraction of the total radiation power falling within the visible range of the spectrum (0.40-0.76 Rm) at the temperature 5000 K; (c) how many times the power radiated at wavelengths exceeding 0.76 Jim will increase if the temperature rises from 3000 to 5000 K.
242
5.259. Making use of Planck's formula, derive the expressions determining the number of photons per 1 cm3of a cavity at a tempe-
rature T in the spectral intervals (0), do) and (X, a,± d2,).
5.260. An isotropic point source emits light with wavelength
= 589 nm. The radiation power of the source is P = 10 W. Find:
(a) the mean density of the flow of photons at a distance r
2.0 m from the source; (b) the distance between the source and the point at which the mean concentration of photons is equal to n = 100 cm -3. 5.261. From the standpoint of the corpuscular theory demonstrate that the momentum transferred by a beam of parallel light rays per unit time does not depend on its spectral composition but de- pends only on the energy flux (De.
5.262. A laser emits a light pulse of duration ti = 0.13 ms and
energy E = 10 J. Find the mean pressure exerted by such a light
pulse when it is focussed into a spot of diameter d = 10 pm on
a surface perpendicular to the beam and possessing a reflection
coefficient p = 0.50.
5.263. A short light pulse of energy E = 7.5 J falls in the form
of a narrow and almost parallel beam on a mirror plate whose reflec- tion coefficient is p = 0.60. The angle of incidence is 30°. In terms of the corpuscular theory find the momentum transferred to the plate.
5.264. A plane light wave of intensity I = 0.20 W/cm2falls on
a plane mirror surface with reflection coefficient p = 0.8. The angle of incidence is 45°. In terms of the corpuscular theory find the magni- tude of the normal pressure exerted by light on that surface.
5.265. A plane light wave of intensity I = 0.70 W/cm2illumi-
nates a sphere with ideal mirror surface. The radius of the sphere is
R = 5.0 cm. From the standpoint of the corpuscular theory find
the force that light exerts on the sphere.
5.266. An isotropic point source of radiation power P is located
on the axis of an ideal mirror plate. The distance between the source and the plate exceeds the radius of the plate n-fold. In terms of the corpuscular theory find the force that light exerts on the plate.
5.267. In a reference frame K a photon of frequency o falls norm-
ally on a mirror approaching it with relativistic velocity V. Find the momentum imparted to the mirror during the reflection of the photon (a) in the reference frame fixed to the mirror;
(b) in the frame K.
5.268. A small ideal mirror of mass m = 10 mg is suspended by
a weightless thread of length 1 = 10 cm. Find the angle through
which the thread will be deflected when a short laser pulse with
energy E = 13 J is shot in the horizontal direction at right angles
to the mirror. Where does the mirror get its kinetic energy? 5.269. A photon of frequency coois emitted from the surface of
a star whose mass is M and radius R. Find the gravitational shift
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