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— Permittivity of subtance sccerding to elementary theory of dispersion:
where nf is the cencentration of electrons of natural frequency ω0.
— Relation betwees refractive indes and permittivity at Bihatance:
n=2
— Phase velocity v and group velocity u:
 v= =/ku=te/dk
— Rayleigh’s, lormula:
v=vλdvdi
— Atlenuation of a narrew beam of electromagnetic ndiation:
I=Ie14
where μ=x+κ,μ,x are the coefficients of linear attenuation, absorption, and wattering.
5.200. A free electron is located in the field of a monochromatic light wave. The intensity of light is I=150 W/m2, its frequency is ω=3.41010 s1. Find:
(a) the electron’s oscillation amplitude and its velocity amplitude:
(b) the ratio F2/Fef  where Fs and F, are  the amplitudes of forces with which the magnetic and electric componeats of the light wave field act on the electron; demonstrate that that ratio is equal to 12v/c, where v is the electron’s velocity amplitude and c is the velocity of light.

Instruetion. The action of the magnetic field component can be disregarded in the equation of motion of the electron since the calculations show it to be negligible.
5.201. An electromagnetic wave of frequency a propagates in dilute plasma. The free electron conceatration is plasma is equal to n0. Neglecting the interaction of the wave and plasma ions, find:
(a) the frequency dependence of plasma permitivity;
(b) how the phase velocity of the electromagnetic wave depends on its wavelength λ in plasma.
5.202. Find the free electron concentration in ionosphere if its refractive index is equal to n=0.90 for radiowaves of frequency v=100MHz.
5.203. Assuming electrons of substance to be free when subjected to hard X-rays, determine by what magnitude the refractive index of graphite diflers from unity in the case of X-tays whose wavelength in vacuum is equal to λ=50pm.
234
5.204. Aa electron experiences a quasi-elastic force kx aad a \»friction force\» γ in the field of electromagnetic radiation. The E-component of the field varies as E=E0cosωt. Neglecting the action of the magnetic componeat of the field, find:
(a) the metion equation of the electron:
(b) the mean power absorbed by the electron; the frequency at which that power is maximum and the expression for the maximum mean power.
5.205. In some cases permittivity of substance turns out to be a complex or a negative quastity, and refractive index, respectively, a complex (n=n+ix) or an imagiary (nix) quantity. Write the equation of a plase wave for beth of these cases and find out the physical meaning of such refractive indices.
5.206. A sounding of dilute plasma by radiowaves of various frequencies reveals that radiowaves with wave. lengths excreding λ0=0.75 m experience total internal reflection. Find the free electron concestration in that plasma.
5.207. Using the definition of the group velocity u, derive Rayleigh’s
Fie. 5.36 . formula (5.5d). Demonstrate that in the viciaity of λ=λ the velocity u is equal to the segment y cut by the tangent of the curve v(λ) at the point λ (Fig. 5.36).
5.208. Find the relation between the group velocity u and phase velocity of for the following dispersion laws:
(a) v is 1/π
(b) v<k;
(c) v<01/ω2.
Here λ,k, asd are the wavelength, wave number, and angular treguency;
5.209. In a certain medium the relationship between the group and phase velocities of an electromagnetic wave has the form uv= =ca, where c is the velocity of light in vacuum. Fiad the dependeace of permittivity of that medium on wave frequency, ε(ω).
5.210. The refractive index of carbon dioxide at the wavelengths 509,334 , and 589 nm is equal to 1.647,1.640, and 1.630 respectively. Calculate the phase aad group velocities of light in the vicinity of λ=534am.
5.211. A train of plane light waves propagates in the medium where the phase velecity v is a linear fuaction of wavelength: v= =a+bλ, where a and b are some positive constants. Demonstrate that in such a medium the shape of as arbitrary train of light waves is restored after the time interval τ=1/b.
5.212. A beam of natural light of intensity I4 falls on a system of two crossed Nicol prisms between which a tube filled with certais
solution is placed in a longitudinal magnetic field of strength H. The length of the tube is i, the coefficient of linear absorption of solution is x, and the Verdet constant is V. Find the inteasity of light transmitted through that system.
5.213. A plane monochromatic light wave of intensity I0 falls normally on a plane-parallel plate both of whose surfaces have a reflection coeficient p. Taking into accoust meltiple reflections, find the intensity of the transmitted light if
(a) the plate is perfectly transpareat, i.e. the absorption is absent;
(b) the coefficient of linear absorption is equal to x, and  the plate thickness is d.
5.214. Two plates, one of thickness d1=3.8 mm and the other of thickness d2=9.0 mm, are manufactured from a certain substance. When placed alternately in the way of monochromatic light, the first transmits τ1=0.84 fraction of lumiaous flux and the second, τ2=0.70. Find the coefficient of linear absorption of that substance. Light falls at right angles to the plates. The secondary reflections are to be aeglected.
5.215. A beam of monochromatie light passes through a pile of N=5 identical plane-parallel glass plates each of thickness l= =0.50 cm. The coefficieat of reflection at each surface of the plates is ρ=0.050. The ratio of the intensity of light transmitted through the pile of plates to the intensity of incident light is τ=0.55. Neglecting the secondary reflections of light, find the absorption coefficieat of the given glass.
5.216. A beam of monochromatic light falls nermally on the surface of a plane-parallel plate of thickaess L. The absorptica coeffcient of the substance the plate is made of varies linearly along the normal to its surface from x1 to x2. The coefficient of reflection at each surface of the plate is equal to p. Neglecting the secondary reflections, find the transmission coefficient of such a plate.
5.217. A beam of light of intensity I0 falls normally on a transparent plane-parallel plate of thickness f. The beam contains all the wavelengths in the interval from λ1 to λ1 of equal spectral intensity. Find the intensity of the transmitted beam if in this wavelength interval the absorption coefficient is a linear function of λ, with extreme values x1 and x2. The coefficient of reflection at each surface is equal to ρ. The secondary reflections are to be aeglected.
5.218. A light filter is a plate of thickness d whose absorption coefficient depends on wavelength λ as
(λ)=a(1λ/λ0)2 cm1
where α and λ2 are constants. Find the passband Δλ of this light filter, that is the band at whose edges the attenuation of light is η times that at the wavelength λ6. The coefficiest of reflection from the surfaces of the light filter is assumed to be the same at all wavejengths.
5.219. A point source of monochromatic light emitting a lumiaous flux Φ is positioned at the centre of a spherical layer of substance. The inside radius of the layer is a, the outside one is b. The coefs. cieat of linear absorption of the substance is equal to %, the reflection coefficient of the surfaces is equal to ρ. Neglecting the secondary reflections, find the inteasity of light that passes through that layer.
5.220. How many times will the intensity of a narrow X-ray beam of wavelength 20pm decrease after passing through a lead plate of thickness d=1.0 mm if the mass absorption coefficient for the given radiation wavelength is equal to μρ=3.6 cm2/g ?
5.221. A aarrow beam of X-ray radiation of wavelength 62 pm penetrates an alumiaium screes 2.6 cm thick. How thick must a lead screea be to attenuate the beam just as much? The mass absorption coefficients of alumiaium and lead for this radiation are equal to 3.48 and 72.0 cm2/g respectively.
5.222. Find the thickness of alumiaium layer which reduces by half the intensity of a narrow monochromatic X-ray beam if the corresponding mass absarption coeflicient is μ/p=0.32 cm2/g.
5.223. How many 50%-absorption layers are there in the plate reducing the intensity of a narrow X-ray beam η=50 times?

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