– Radiesity
\[
M_{4}=\frac{\epsilon}{6}=
\]
where o is the space density of thermal radiation enercy.
– Wien’s formala and Wien’s displacement law:
where $\lambda_{n}$ is the wavelength eorresponding to the maxiaus of the fusction $w_{2}$.
– Stefon-Boltumans law:
\[
M_{*}=a r,
\]
where $\sigma$ is the Stefan-Boltzmann constant.
– Planek’s formula:
– Einstein’s pheteelectric equation:
\[
\text { Ao- } A+\frac{\operatorname{man}^{2}}{2} \text {. }
\]
94
94
\[
\sin -A+\frac{m=x}{2}
\]

5.253. A cavity of volume $V=1.01$ is flled with thermal radiation at a temperature $T=1000 \mathrm{~K}$. Find:
(a) the heat capacity $\bar{C}_{v}$; (b) the entropy $S$ of that radiation.
5.254. Assuming the spectral distribution of thermal radiation energy to obey Wien’s formula $u(\omega, T)=A \omega^{2} \exp (-\epsilon \omega / T)$, where $a=7.64 \mathrm{ps} \cdot \mathrm{K}$, find for a temperature $T=2000 \mathrm{~K}$ the most probable
(a) radiation frequency; (b) radiation wavelength.
5.255. Using Planck’s formula, derive the approximate expressions for the space spectral density us of radiation
(a) in the range where $A_{\omega} \ll \bar{k} T$ (Rayleigh-Jeans formula);
(b) in the range where $A_{\omega}>k T$ (Wien’s formula).
5.256. Transform Planck’s formula for space spectral density us of radiation from the variable o to the variables $v$ (linear frequency) and $\lambda$ (wavelength).
5.257. Using Planck’s formula, find the power radiated by a unit area of a black body within a narrow wavelength interval $\Delta \lambda$ $=1.0 \mathrm{~nm}$ close to the maximum of spectral radiation density at a temperature $T=3000 \mathrm{~K}$ 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
the spectral interval from 0 to $x$. Here $x=\lambda / \lambda_{n} a_{n}$ is the wavelength corresponding to the maximum of spectral radiation density).
Using this plet, find:
(a) the wavelength which divides the radiation spectrum into two equal (in terms of energy) parts at the temperature $3700 \mathrm{~K}$ :
(b) the fraction of the total radiation power falling within the visible range of the spectrum $(0.40-0.76 \mu \mathrm{m})$ at the temperature so00 K:
(c) how many times the power radiated at wavelengths exceeding $0.76 \mu \mathrm{m}$ will increase if the temperature rises from 3000 to $5000 \mathrm{~K}$.
5.259. Making use of Planck’s formula, derive the expressions determining the number of photens per $1 \mathrm{~cm}^{2}$ of a cavity at a temperature $T$ in the spectral intervals $(\omega, \omega+d \omega)$ and $(\lambda, \lambda+\lambda)$.
5.260. An isotropic point source emits light with wavelength $\lambda=589 \mathrm{~nm}$. The radiation power of the source is $P=10 \mathrm{~W}$. Find:
(a) the mean density of the flow of photons at a distance $r=$ $=2.0 \mathrm{~m}$ from the source;
(b) the distance between the source and the point at which the mean concentration of photons is equal to $n=100 \mathrm{~cm}^{3}$.
5.261. From the stand point of the corpuscular theory demonstrate that the mementum transferred by a beam of parallel light rays per unit time does not depend on its spectral composition but depends oaly on the energy flux $\Phi_{r}$ –
5.262. A laser emits a light pulse of duration $\tau-0.13 \mathrm{~ms}$ and energy $E=10 \mathrm{~J}$. Find the mean pressure exerted by such a light pulse when it is focussed into a spot of diameter $d=10 \mu \mathrm{m}$ on a surface perpendicular to the beam and possessing a reflection coefficient $\rho=0.50$.
5.263. A short light pulse of energy $E=7.5 \mathrm{~J}$ falls in the form of a narrow and almost parallel beam on a mirror plate whose reflection coefficieat is $\rho=0.60$. The angle of incidence is $30^{\circ}$. In terms of the corpuscular theory find the momeatum translerred to the plate.
5.264. A plane light wave of intensity $I=0.20 \mathrm{~W} / \mathrm{cm}^{\prime}$ falls on a plane mirror surface with reflection coefficient $p=0.8$. The angle of incidence is $45^{\circ}$, In terms of the corpuscular theory fiad the magnitude of the normal pressure exerted by light on that surface.
5.265. A plane light wave of intensity $I=0.70 \mathrm{~W} / \mathrm{cm}^{2}$ illuminates a sphere with ideal mirror surface. The radius of the sphere is $\boldsymbol{A}=5.0 \mathrm{~cm}$. From the stand point 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 $\eta$-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 a flals normally on a mirror appraching it with relativistic velocity $\boldsymbol{V}$. Find the momentum imparted to the mirror during the reflection of the photen
(a) in the reference frame fixed to the mirror;
(b) is the frame $K$.
5.268. A small ideal mirrer of mass $m=10 \mathrm{mg}$ is suspended by – weightless thread of length $l=10 \mathrm{~cm}$. Find the angle through which the thread will be deflected whea a short laser pulse with energy $E=13 \mathrm{~J}$ is shot in the horizeatal direction at right angles to the mirror. Where does the mirror get its kinetic energy?
5.269. A photon of frequency $\omega$, is emitted from the surface of a star whose mass is $M$ and radius $R$. Find the gravitational shift
of frequency $\Delta_{\omega} / \omega_{0}$ of the photon at a very great distance from the star.
5.270. A voltage applied to an $X$-ray tube being increased $\eta_{-}-$ $=1.5$ times, the short-wave limit of an $X$-ray continuous spectrum shifts by $\Delta \mathbf{\lambda}-26 \mathrm{pm}$. Find the initial voltage applied to the tube.
5.271. A narrow $\mathrm{X}$-ray beam falls on a $\mathrm{NaCl}$ single crystal. The least angle of incidence at which the mirror reflection from the system of crystallographic planes is still observed is equal to $a=$ -4.1\”. The interplanar distance is $d=0.28 \mathrm{~mm}$. How high is the voltage applied to the $\mathrm{X}$-ray tube?
5.272. Find the wavelength of the short-wave limit of an X-ray continuous spectrum if electrons approach the anticathode of the tube with velocity $v=0.85 c$, where $c$ is the velocity of light.
5.273. Find the photoelectric threshold for zinc and the maximum velocity of photoelectrons liberated from its surface by electromagnetie radiation with wavelength $250 \mathrm{~nm}$.
5.274. Illuminating the surface of a certain metal alternately with light of wavelengths $\lambda_{1}=0.35 \mu \mathrm{m}$ and $\lambda_{7}=0.54 \mu \mathrm{m}$. it was found that the corresponding maximum velocities of photoelectrons differ by a factor $\eta=2.0$. Find the work function of that metal.
5.275. Up to what maximum potential will a cepper ball, remote from all other bodies, be charged when irradiated by electromagnetic radiation of wavelength $\lambda=140 \mathrm{nz}$ ?
5.276. Find the maximum kinetic energy of photoelectrons liberated from the surface of lithium by electromagnetic radiation whose electric component varies with time as $E=a(1+\cos \omega t) \cos \omega_{0} t$. where $a$ is a constant, $\omega=6.0 \cdot 10^{14} \mathrm{~s}^{-1}$ and $\Theta_{0}=3.60 \cdot 10^{15} \mathrm{~s}^{-1}$.
5.277. Electromagnetic radiation of wavelength $\lambda=0.30 \mu \mathrm{m}$ falls on a photocell operating in the saturation mode. The corresponding spectral sensitivity of the photocell is $J=4.8 \mathrm{~m} / \mathrm{W}$. Find the yield of photeelectrons, i.e. the number of photeelectrons produced by each incident photon.
5.278 . There is a vacuum photocell whose ene electrode is made of cesium and the other of copper. Find the maximun velocity of photoelectrons approaching the copper electrode when the cesium electrode is subjected to electromagnetic radiation of wavelength $0.22 \mu \mathrm{m}$ and the electrodes are shorted outside the cell.
5.279 . A photoelectric current emerging in the circuit of a vacuum photocell when its zinc electrode is subjected to electromagnetic radiation of wavelength $262 \mathrm{~nm}$ is cancelled if an external decelerating voltage $1.5 \mathrm{~V}$ is applied. Find the magnitude and polarity of the outer contact potential difference of the given photocell.
5.280. Compose the expression for a quantity whose dimension is length, using velocity of light $e$, mass of a particle $m$, and Planck’s constant $\boldsymbol{n}$. What is that quantity?
5.381. Using the conservation laws, demonstrate that a free electron cannot absorb a photon completely.
264
5.282. Explain the following features of Compton scattering of light by matter:
(a) the increase in wavelength $\Delta \boldsymbol{\lambda}$ is independent of the nature of the scattering substance:
(b) the intensity of the displaced component of scattered light grows with the increasing angle of scattering and with the diminish. ing atemic number of the substance;
(c) the presence of a non-displaced component in the scattered radiation.
5.283. A narrow monochromatic $\mathrm{X}$-ray beam falls on a scattering substance. The wavelengths of radiation scattered at angles $\theta_{1}=60^{\circ}$ and $\theta_{2}=120^{\circ}$ differ by a factor $\eta=2.0$. Assuming the free electrons to be responsible for the scattering, find the incident radiation wavelength.
5.284. A photon with energy $h \in=1.00 \mathrm{MeV}$ is seattered by a stationary free electron. Find the kinetic energy of a Compten electron if the photon’s wavelength changed by $\eta=25 \%$ due to scattering.
5.285. A photon of wavelength $\lambda=6.0 \mathrm{pm}$ is scattered at right angles by a stationary free electron. Find:
(a) the frequency of the scattered photon;
(b) the kinetic energy of the Compton electron.
5.286. A photen with energy $A \omega-250 \mathrm{keV}$ is scattered at on angle $6-120^{\circ}$ by a stationary free electron. Find the energy of the scattered photon.
5.287. A photon with momentum $p=1.02 \mathrm{MeV} / e$, where $e$ is the velocity of light, is scattered by a stationary free electron, changing in the process its momentum to the value $p^{\prime}=0.255 \mathrm{MeV} / \mathrm{e}$. At what angle is the photon scattered?
5.288. A photon is scattered at an angle $\theta-120^{\circ}$ by a stationary free electron. As a result, the electron acquires a kinetic energy $T=0.45 \mathrm{MeV}$. Find the energy that the photon had prior to scattering.
5.289. Find the wavelength of X-ray radiation if the maximum kinetic energy of Compton electrons is $T_{=0 y}=0.19 \mathrm{MeV}$.
5.290. A photen with energy $\mathrm{A}_{\omega}=0.15 \mathrm{MeV}$ is scattered by a stationary free electron changing its wavelength by $\mathbf{\Delta \lambda}=\mathbf{3 . 0} \mathrm{pm}$. Find the angle at which the Compton electron moves.
5.291. A photon with energy exceeding $\eta=2.0$ times the rest energy of an electron experienced a head-on collision with a stationary free electron. Find the curvature radius of the trajectory of the Compton electron in a magnetic field $B=0.12 T$. The Compton electron is assumed to move at right angles to the direction of the feld.
5.292. Having collided with a relativistic electron, a photon is deflected through an angle $\theta=60^{\circ}$ while the electron stops. Find the Compton displacement of the wavelength of the scattered photon.