– Permittivity of subtance sccerding to elementary theory of dispersion:
where $\mathrm{n}_{\mathrm{f}}$ is the cencentration of electrons of natural frequency $\omega_{0}$.
– Relation betwees refractive indes and permittivity at Bihatance:
\[
n=\sqrt{2} \text {. }
\]
– Phase velocity $v$ and group velocity u:
\[
\text { v= }=/ k \text {, } u=t e / d k \text {. }
\]
– Rayleigh’s, lormula:
\[
v=v-\lambda \frac{d v}{d i} \text {. }
\]
– Atlenuation of a narrew beam of electromagnetic ndiation:
\[
I=I e^{-14} \text {, }
\]
where $\mu=x+\kappa^{\prime}, \mu, x^{\prime}$ 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 $I=150 \mathrm{~W} / \mathrm{m}^{2}$, its frequency is $\omega=3.4 \cdot 10^{10} \mathrm{~s}^{-1}$. Find:
(a) the electron’s oscillation amplitude and its velocity amplitude:
(b) the ratio $F_{2} / F_{\text {ef }}$ where $F_{\mathrm{s}}$ and $F_{\text {, are }}$ 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 $\frac{1}{2} v / c$, 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 $n_{0}$. 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 $\lambda$ 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 $\mathrm{v}=100 \mathrm{MHz}$.
5.203. Assuming electrons of substance to be free when subjected to hard $\mathrm{X}$-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 $\lambda=50 \mathrm{pm}$.
234
5.204. Aa electron experiences a quasi-elastic force $k x$ aad a \”friction force\” $\gamma^{*}$ in the field of electromagnetic radiation. The $E$-component of the field varies as $E=E_{0} \cos \omega t$. 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 $\left(n^{\prime}=n+i x\right)$ or an imagiary $\left(n^{\prime}-i x\right)$ 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 $\lambda_{0}=0.75$ 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 $\lambda=\lambda^{*}$ the velocity $u$ is equal to the segment $y^{\prime}$ cut by the tangent of the curve $v(\lambda)$ at the point $\lambda^{\prime}$ (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 $1 / \sqrt{\pi}$
(b) $v<k$;
(c) $v<01 / \omega^{2}$.
Here $\lambda, k$, 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 $u v=$ $=c^{a}$, where $c$ is the velocity of light in vacuum. Fiad the dependeace of permittivity of that medium on wave frequency, $\boldsymbol{\varepsilon}(\omega)$.
5.210. The refractive index of carbon dioxide at the wavelengths 509,334 , and $589 \mathrm{~nm}$ is equal to $1.647,1.640$, and 1.630 respectively. Calculate the phase aad group velocities of light in the vicinity of $\lambda=534 \mathrm{am}$.
5.211. A train of plane light waves propagates in the medium where the phase velecity $v$ is a linear fuaction of wavelength: $v=$ $=a+b \lambda$, 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 $\tau=1 / b$.
5.212. A beam of natural light of intensity $I_{4}$ 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 $I_{0}$ 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 $x_{\text {, and }}$ the plate thickness is $d$.
5.214. Two plates, one of thickness $d_{1}=3.8 \mathrm{~mm}$ and the other of thickness $d_{2}=9.0 \mathrm{~mm}$, are manufactured from a certain substance. When placed alternately in the way of monochromatic light, the first transmits $\tau_{1}=0.84$ fraction of lumiaous flux and the second, $\tau_{2}=0.70$. 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 $l=$ $=0.50 \mathrm{~cm}$. The coefficieat of reflection at each surface of the plates is $\rho=0.050$. The ratio of the intensity of light transmitted through the pile of plates to the intensity of incident light is $\tau=0.55$. 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 $x_{1}$ to $x_{2}$. 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 $I_{0}$ falls normally on a transparent plane-parallel plate of thickness $f$. The beam contains all the wavelengths in the interval from $\lambda_{1}$ to $\lambda_{1}$ of equal spectral intensity. Find the intensity of the transmitted beam if in this wavelength interval the absorption coefficient is a linear function of $\lambda$, with extreme values $x_{1}$ and $x_{2}$. The coefficient of reflection at each surface is equal to $\rho$. The secondary reflections are to be aeglected.
5.218. A light filter is a plate of thickness $d$ whose absorption coefficient depends on wavelength $\lambda$ as
\[
*(\lambda)=a\left(1-\lambda / \lambda_{0}\right)^{2} \mathrm{~cm}^{-1} \text {, }
\]
where $\alpha$ and $\lambda_{2}$ are constants. Find the passband $\Delta \lambda$ of this light filter, that is the band at whose edges the attenuation of light is $\eta$ times that at the wavelength $\lambda_{6}$. 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 $\Phi$ 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 $\rho$. 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 $\mathrm{X}$-ray beam of wavelength $20 \mathrm{pm}$ decrease after passing through a lead plate of thickness $d=1.0 \mathrm{~mm}$ if the mass absorption coefficient for the given radiation wavelength is equal to $\mu_{\rho}=3.6 \mathrm{~cm}^{2} / \mathrm{g}$ ?
5.221. A aarrow beam of $X$-ray radiation of wavelength 62 pm penetrates an alumiaium screes $2.6 \mathrm{~cm}$ 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 $72.0 \mathrm{~cm}^{2} / \mathrm{g}$ respectively.
5.222. Find the thickness of alumiaium layer which reduces by half the intensity of a narrow monochromatic $\mathrm{X}$-ray beam if the corresponding mass absarption coeflicient is $\mu / p=0.32 \mathrm{~cm}^{2} / \mathrm{g}$.
5.223. How many $50 \%$-absorption layers are there in the plate reducing the intensity of a narrow $X$-ray beam $\eta=50$ times?