Problems in General Physics → Optics → Thermal Radiation. Quantum Nature of Light
5.247. The temperature of one of the two heated black bodies is T1 = 2500 K. Find the temperature of the other body if the wavelength corresponding to its maximum emissive capacity exceeds by Δλ = 0.50 μm the wavelength corresponding to the maximum emissive capacity of the first black body.
5.248. The radiosity of a black body is Me = 3.0 W/cm2. Find the wavelength corresponding to the maximum emissive capacity of that body.
5.249. The spectral composition of solar radiation is much the same as that of a black body whose maximum emission corresponds to the wavelength 0.48 μm. Find the mass lost by the Sun every second due to radiation. Evaluate the time interval during which the mass of the Sun diminishes by 1 per cent.
5.250. Find the temperature of totally ionized hydrogen plasma of density ρ = 0.10 g/cm3 at which the thermal radiation pressure is equal to the gas kinetic pressure of the particles of plasma. Take into account that the thermal radiation pressure p = u/3, where u is the space density of radiation energy, and at high temperatures all substances obey the equation of state of an ideal gas.
5.251. A copper ball of diameter d = 1.2 cm was placed in an evacuated vessel whose walls are kept at the absolute zero temperature. The initial temperature of the ball is T0 = 300 K. Assuming the surface of the ball to be absolutely black, find how soon its temperature decreases η = 2.0 times.
5.252. There are two cavities (Fig. 5.39) with small holes of equal diameters d = 1.0 cm and perfectly reflecting outer surfaces. The distance between the holes is l = 10 cm. A constant temperature T1 = 1700 K is maintained in cavity 1. Calculate the steady-state temperature inside cavity 2.
5.253. A cavity of volume V = 1.0 l is filled with thermal radiation at a temperature T = 1000 K. Find:
5.255. Using Planck's formula, derive the approximate expressions for the space spectral density uω of radiation
5.257. Using Planck's formula, find the power radiated by a unit area of a black body within a narrow wavelength interval Δλ = 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 the spectral interval from 0 to x. Here x = λ/λm (λm is the wavelength corresponding to the maximum of spectral radiation density).
5.260. An isotropic point source emits light with wavelength λ = 589 nm. The radiation power of the source is P = 10 W. Find:
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 depends only on the energy flux Φe.
5.262. A laser emits a light pulse of duration τ = 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 μm on a surface perpendicular to the beam and possessing a reflection coefficient ρ = 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 reflection coefficient is ρ = 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/cm2 falls on a plane mirror surface with reflection coefficient ρ = 0.8. The angle of incidence is 45°. In terms of the corpuscular theory find the magnitude of the normal pressure exerted by light on that surface.
5.265. A plane light wave of intensity I = 0.70 W/cm2 illuminates 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.
5.267. In a reference frame K a photon of frequency ω falls normally on a mirror approaching it with relativistic velocity V. Find the momentum imparted to the mirror during the reflection of the photon
5.270. A voltage applied to an X-ray tube being increased η = 1.5 times, the short-wave limit of an X-ray continuous spectrum shifts by Δλ = 26 pm. Find the initial voltage applied to the 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 electromagnetic radiation with wavelength 250 nm.
5.274. Illuminating the surface of a certain metal alternately with light of wavelengths λ1 = 0.35 μm and λ2 = 0.54 μm, it was found that the corresponding maximum velocities of photoelectrons differ by a factor ν = 2.0. Find the work function of that metal.
5.275. Up to what maximum potential will a copper ball, remote from all other bodies, be charged when irradiated by electromagnetic radiation of wavelength λ = 140 nm?
5.277. Electromagnetic radiation of wavelength λ = 0.30 μm falls on a photocell operating in the saturation mode. The corresponding spectral sensitivity of the photocell is J = 4.8 mA/W. Find the yield of photoelectrons, i.e. the number of photoelectrons produced by each incident photon.
5.278. There is a vacuum photocell whose one electrode is made of cesium and the other of copper. Find the maximum velocity of photoelectrons approaching the copper electrode when the cesium electrode is subjected to electromagnetic radiation of wavelength 0.22 μm and the electrodes are shorted outside the cell.
5.283. A narrow monochromatic X-ray beam falls on a scattering substance. The wavelengths of radiation scattered at angles θ1 = 60° and θ2 = 120° differ by a factor η = 2.0. Assuming the free electrons to be responsible for the scattering, find the incident radiation wavelength.
5.284. A photon with energy ħω = 1.00 MeV is scattered by a stationary free electron. Find the kinetic energy of a Compton electron if the photon's wavelength changed by η = 25% due to scattering.
5.285. A photon of wavelength λ = 6.0 pm is scattered at right angles by a stationary free electron. Find:
5.286. A photon with energy ħω = 250 keV is scattered at an angle θ = 120° by a stationary free electron. Find the energy of the scattered photon.
5.287. A photon with momentum p = 1.02 MeV/c, where c is the velocity of light, is scattered by a stationary free electron, changing in the process its momentum to the value p' = 0.255 MeV/c. At what angle is the photon scattered?
5.289. Find the wavelength of X-ray radiation if the maximum kinetic energy of Compton electrons is Tmax = 0.19 MeV.
5.290. A photon with energy ħω = 0.15 MeV is scattered by a stationary free electron changing its wavelength by Δλ = 3.0 pm. Find the angle at which the Compton electron moves.
5.291. A photon with energy exceeding η = 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 field.