– Degree of polarisatios of light:
\[
P=\frac{I_{\max }-I_{\operatorname{mn}}}{T_{\max }+I_{\min }} .
\]
– Malus’, law:
\[
I=I, \cos ^{2} 4 .
\]
– Brewster’s law:
\[
\tan \theta_{n}=n, / n_{p} \text {. }
\]
– Fresel equations for intensity of light reflected at the boundary betwees two dielecirics:
\[
r_{1}^{\prime}=I_{2} \frac{\sin ^{2}\left(\theta_{1}-\theta_{2}\right)}{\sin ^{2}\left(\theta_{1}+\theta_{2}\right)}, \quad r_{1}=I_{1} \frac{\tan ^{2}\left(\theta_{1}-\theta_{2}\right)}{\tan ^{2}\left(\hat{\beta}_{2}+\theta_{2}\right)},
\]
where $I_{4}$ and $I_{n}$ are the intensities of ineident light whose electrie vecter oscil. lations are respectively perpendicular asd paraflet to the plane of incidence. – A crystalline plate between two polariaers $P$ and $P^{\prime}$, if the angle betwest the intensity $F$ ‘ of light which passes throush the polarizer $P$ ‘ turns out to be either maximum or minimule under the fliowing cosditions:

Here $\delta=2 x\left(n_{0}-n_{q}\right) d \lambda$ is the phase differebce betwees the ordisary and
– Natural asd magnetic rotation of the plase of polariation:
where $a$ is the rotation constant, $\boldsymbol{V}$ is Verdet’s cosstant.
5.157. A plane mosochromatic wave of natural light with inten: sity $I$, falls normally on a screen composed of two touching Polaroid half-planes. The principal direction of one Polaroid is parallel,
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and of the other perpeadicular, to the bouadary between them. What kind of diffraction pattern is formed behind the screen? What is the intensity of light behind the screen at the points of the plase perpendicular to the screen and passing through the boundary between the Polaroids?
5.158. A plane monochromatic wave of atural light with intensity $I$, falls normally on an opaque screen with round hole corresponding to the first Fresnel zone for the observation point P. Find the intensity of light at the point $P$ after the hole was covered with two idestical Polaroids whose priacipal directions are mutually perpendicular and the boundary between them passes
(a) along the diameter of the hole;
(b) along the circumference of the circle limiting the first half of the Fresiel zone.
5.159. A beam of plane-polarized light falls on a polarizer which rotates about the axis of the ray with angular velocity $\omega=21 \mathrm{rad} / \mathrm{s}$. Find the energy of light passing through the polarizer per one revolution if the fiux of energy of the incident ray is equal to $\Phi_{0}=$ $=4.0 \mathrm{~mW}$.
5.160. A beam of natural light falls on asystem oi $N=6$ Nicol prisms whose transmission planes are turned each through an angle $\varphi=30^{\circ}$ with respect to that of the foregoing prism. What fraction of lumiaous fux passes through this system?
5.161. Natural light falls on a system of three identical is-line Polarofds, the principal direction of the middle Polaroid forming an angle $\varsigma=60^{\circ}$ with those of two other Polaroids. The maximum transmission coefficient of each Polaroid is equal to $\mathrm{r}=0.81$ when plane-polarized light falls on them. How many times will the intensity of the light decrease after its passing through the system?
5.162. The degree of polarization of partially polarized light is $P=0.25$. Find the ratio of intensities of the polarized component of this light and the aatural component.
5.463. A Nicol prism is placed in the way of partially polarised besm of light. When the prism is turned from the position of maximum transmission through an angle $q=60^{\circ}$, the intensity of transmitted light decreased by a factor of $\eta=3.0$. Find the degree of polarization of incident light.
5.164. Two identical imperfect polarizers are placed in the way of a natural beam of light. When the polarizers’ planes are parallel, the system trasmits $\eta=10,0$ times more light than in the case of crossed planes. Find the degree of polarization of light produced
(a) by each polarizer separately:
(b) by the whole system when the plases of the polarizers are parallel.
5.165. Two parallel plane-pelarized beams of light of equal intensity whose oscillation planes $N_{1}$ and $N_{1}$ form a small angle $\uparrow$ between
5.189. Using the tables of the Appendix, calculate the difference of refractive indices of quarta for light of wavelength $\lambda=589.5 \mathrm{~nm}$ with right-hand and left-hand circular polarizations.
5.190. Plane-polarized light of wavelength $0.59 \mu \mathrm{m}$ falls on a trihedral quarts prism $P$ (Fig. 5.34) with refrecting angle $\theta=$ $=30^{\circ}$. Inside the prism light propagates along the optical axis whose direction is shows by hatehing. Behind the Polaroid Pol an interlerence pattern of bright and dark fringes of width $\Delta x=$ $=15.0 \mathrm{~mm}$ is observed. Find the specific rotation constant of quarts and the distribution of intensity of light behind the Polaroid.
5.191. Natural monochromatic light falls on a system of two erossed Nicol prisms between which a quarts plate cut at right angles to its optical axis is inserted. Find the minimum thickness of the plate at which this system will trassmit $\eta=0.30$ of lumisons Aux if the specific rotation constant of quarti is equal to $\alpha=17$ ang.deg $/ \mathrm{mm}$.
5.192. Light passes through a system of two chossed Nicol prisms between which a quarte plate cet at right angles to its optical axis is placed. Determine the minimum thickness of the plate which allows light of wavelength $436 \mathrm{~nm}$ to be completely cut off by the system and transmits half the light of wavelength $497 \mathrm{~nm}$. The specific rotation constant of quartz for these wavelengths is equal to 41.5 and 31.1 angular degrees per $m m$ respectively.
5.193. Plane-polarized light of wavelength $589 \mathrm{~nm}$ propagates along the axis of a cylindrical glass vessel filled with slightly turbid sugar solution of concentration $500 \mathrm{~g} /$. Viewing from the side, one can see a system of helical fringes, with $50 \mathrm{~cm}$ between neighbouring dark fringes along the axis. Explain the emergence of the fringes and determine the specific rotation constant of the solution.
5.194. A Kerr cell is positioned between two erosed Nicol prisms so that the direction of electric field $\mathbf{E}$ in the capacitor forms an angle of $45^{\circ}$ with the principal directions of the prisms. The capacitor has the length $t=10 \mathrm{~cm}$ and is filled up with aitrobenzene. Light of wavelength $\lambda=0.50 \mu \mathrm{m}$ passes through the system. Taking into account that in this case the Kerr constant is equal to $B=$ $=2.2 \cdot 10^{-10} \mathrm{~cm} / \mathrm{V}^{i}$, find:
(a) the minimum strength of electric field $E$ in the capacitor at which the intensity of light that passes through this system is independent of rotation of the rear prism:
(b) how many times per second light will be interrupted when a sinusoidal voltage of frequency $v=10 \mathrm{MHz}$ and strength amplitude $E_{m}=50 \mathrm{kV} / \mathrm{cm}$ is applied to the capaciter.

Note. The Kerr constant is the coefficient $B$ in the equation $n_{q}-$ $-n_{0}=B \lambda E^{2}$.
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5.195. Monochromatic plane-polarized light with angular frequency 0 passes through a certain substance along a uniform magnetic feld $H$. Find the difference of refractive indices for right-hand and left-hand components of light beam with circular polarization if the Verdet constant is equal to $V$.
5.196. A certain substance is placed in a longitudinal magnetic field of a solesoid located between two Polaroids. The length of the tube with substance is equal to $t=30 \mathrm{~cm}$. Find the Verdet constant if at a field strength $H=56.5 \mathrm{kA} / \mathrm{m}$ the angle of rotation of polarization plane is equal to $\%_{1}=+5^{\circ} 10^{\prime}$ for one direction of the field and to $\varphi_{1}=-3^{\circ} 20^{\prime}$, for the opposite direction.
5.197. A narrow beam of plane-polarized light passes through dextrorotatory positive compound placed into a longitudinal magnetie field as shown in Fig. 5.35. Find the angle through which the
Fig. 5.35.
polarization plase of the transmitted beam will turn if the length of the tube with the compouad is equal to $b$, the specific rotation constant of the compound is equal to $\boldsymbol{a}$, the Verdet constent is $\boldsymbol{y}$, and the magnetic field strength is $H$.
5.198. A tube of length $l=26 \mathrm{~cm}$ is filled with benzene and placed is a longitudinal magnetic field of a solenoid positioned between two Polaroids. The angle between the prisciple directions of the Polaroids is equal to $45^{\circ}$. Find the minimum strength of the magnetie field at which light of the wavelength $589 \mathrm{~nm}$ propagates through that system only in one direction (optical valve). What happens if the direction of the given magnetic field is changed to the opposite oae?
5.199. Experience shows that a body irradiated with light with circular polarization acquires a torque. This happens because such a light possesses an angular momentum whose hlow deasity in vacuum is equal to $M=1 / 0$, where $I$ is the intensity of light, 0 is the angular oscillation frequeacy. Suppose light with circular polarization and wavelength $\lambda=0.70 \mu \mathrm{m}$ falls normally on a aniform black dise of mass $m=10 \mathrm{mg}$ which cas freely rotate about its axis. How soon will its angular velocity become equal to $\omega_{0}=$ $=1.0 \mathrm{rad} / \mathrm{s}$ provided $I=10 \mathrm{~W} / \mathrm{cm}^{2}$ ?