– Width of a fring:
\[
\Delta x=\frac{t}{d} \lambda
\]
where $t$ is the distance from the sources to the scren, $d$ is the distance between the rources.
– Temporal and spatial coherences. Coherence lenglh and coherence radius:
\[
l_{\operatorname{eat}}=\frac{\lambda v}{\Delta x}, P_{\text {rat }} \sim \frac{\lambda}{8},
\]
where $\$$ is the angular dimension of the source.
– Condition lor interference maxima is the case of light reflected from a this plate of thickness b:
\[
2 b \sqrt{n^{2}-\sin ^{2} \sigma_{1}}=(k+1 / 2) \lambda,
\]
where $k$ is an integer.
– Newtos’s ring produced on reflection of light from the surfaces of as the conver surlace of the lese is in contact. The radii at the ning:
\[
r=\sqrt{\text { 2ाk/2, }}
\]
with the rines being bright if $k=1,3,5, \ldots$, and dark if $k-2,4,6, \ldots$ The value $k=0$ corresponds to the iniddle of the central dark spot.
5.64. Demonstrate that when two harmonic oscillations are added, the time-averaged energy of the resultant oscillation is equal to the sum of the energies of the constituent oscillations, if both of them
(a) have the same direction and are incoherent, and all the values of the phase diflerence between the escillations are equally probable;
(b) are mutually perpendicular, have the same frequency and an arbitrary phase difference.
5.65. By means of plotting find the amplitude of the oscillation resulting from the addition of the following three ascillations of the same direction:
5.66. A certain oscillation results from the addition of coherent oscillations of the same direction $\mathrm{s}=a \cos \mathrm{l} \omega t+(k-1)$ vl. where $k$ is the number of the oscillation $(k=1,2, \ldots, N), \stackrel{q}{*}$ is the phase difference between the $k$ th and $(k-1)$ th escillations. Find the amplitude of the resultant escillation.
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5.67. A system illustrated in Fig. 5.12 censists of two coherent point sources $I$ and 2 located in a certain piane so that their dipole moments are oriented at right angles to that plane. The sources are separated by a distance d, the radiation wavelength is equal to $\lambda$. Taking inte account that the oscillations of source 2 lag in phase behind the oscillations of souree $I$ by $\%(\varsigma<\pi)$, find:
(a) the angles $\theta$ at which the radiation intensity is maximam;
(b) the conditions under which the radiation intensity in the direction $\theta=n$ is maximum and in the epposite direction, minimum.
5.68. A stationary radiating system consists of a linear chain of parallel oscillaters separated by a distance $d$, with the oscillation phase varying linearly along the chain. Find the time dependence of the phase diflerence $\Delta y$ between the neighbouring oscillaters at which the principal radiation maximum of the system will be \”scanning\” the surroundings with the constant angular velocity $\Leftrightarrow$.
5.69. In Lloyd’s mirror experiment (Fig. 5.13) a light wave emitted directly by the source $S$ (narrow slit) interferes with the wave reflected from a mirror $M$. As a result, an interfereace fringe pattern is
Fig. S.13.
formed on the screen $S c$. The source and the mirror are separated by a distance $l=100 \mathrm{~cm}$. At a certain position of the source the fringe width on the screen was equal to $\Delta \boldsymbol{x}=\mathbf{0 . 2 5} \mathrm{mm}$, and after the soure was meved away from the mirror plane by $\Delta h=0.60 \mathrm{~mm}$, the friage width decreased $\eta=1.5$ times. Find the wavelength of light.
5.70. Two ceherent plane light waves propagating with a diverzence angle \& $\boldsymbol{1}$ fall almost nermally on a screen. The amplitudes of the waves are equal. Demonstrate that the distance between the neighbouring maxima on the screen is equal to $\Delta x=\lambda / 4$, where $\lambda$ is the wavelength.
5.71. Figure 5.14 illustrates the interference experiment with Fresnel mirrors. The anigle between the mirrors is $\alpha=12$, the distances frem the mirror’ intersection line to the narrew stit $S$ and the screen $S c$ are equal to $r=10.0 \mathrm{~cm}$ and $b=130 \mathrm{~cm}$ respectively. The wavelength of light is $\lambda=0.55 \mu \mathrm{m}$. Find:
(a) the width of a fringe on the sereen and the number of possible maxima;
(b) the shift of the interference pattern on the sereen when the slit is displaced by $8 l=1.0 \mathrm{~mm}$ along the are of radius $r$ with centre at the point $O$;
(c) at what maximum width $\delta_{\text {mox }}$ of the slit the interference fringes on the screen are still observed sufficiently sharp.
Fie. 5.14.
5.72. A plane light wave falls on Fresnel mirror with an angle $\alpha=2.0^{\circ}$ between them. Determine the wavelength of light if the width of the fringe on the sereen $\Delta x=0.55 \mathrm{~mm}$.
5.73. A leas of diameter $5.0 \mathrm{~cm}$ and focal length $f=25.0 \mathrm{~cm}$ was cut aleng the diameter into two identical halves. In the process, the layer of the lens $a=1.00 \mathrm{~mm}$ in thickness was lost. Then the halves were put together to form a composite lens. In this focal plane a narrow slit was placed, emitting menochromatic light with wavelength $\lambda=0.60 \mu \mathrm{m}$. Behind the lens a screen was located at a distance $b=50 \mathrm{~cm}$ from it. Find:
(a) the width of a fringe on the sereen and the number of possible maxima;
(b) the maximum width of the slit $\delta_{\mathrm{mex}}$ at which the fringes on the screen will be still observed sufficiently sharp.
5.74. The distances from a Fresnel biprism to a narrow slit and a sereen are equal to $a=25 \mathrm{~cm}$ and $b=100 \mathrm{~cm}$ respectively. The refracting angle of the glass biprism is equal to $\theta=20^{\circ}$. Find the wavelength of light if the width of the fringe on the screen is $\Delta x=0.55 \mathrm{~mm}$.
5.75. A plane light wave with wavelength $\lambda=0.70 \mu \mathrm{m}$ falls normally on the base of a biprism made of glass $(n-1.520)$ with refracting angle $\theta=$ $=5.0^{\circ}$. Behind the biprism (Fig. 5.15)

Fie. 5.15. there is a plase-parallel plate, with the space between them filled up with benzene $\left(n^{*}-1.500\right)$. Find the width of a fringe on the serven Se placed behind this system.
5.76. A plane monochromatic light wave falls normally on a diaphragm with two narrow slits separated by a distance $d=2.5 \mathrm{~mm}$.
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A fringe pattern is formed on a screen placed at a distance $t=$ $=100 \mathrm{~cm}$ behind the diaphragm. By what distance and in which direction will these fringes be displaced when one of the stits is covered by a glass plate of thickness $h=10 \mu \mathrm{m}$ ?
5.77. Figure 5.16 illustrates an interferometer used in measurements of refractive indices of transparent substances. Here $S$ is
Fie. S.16.
a sarrow slit illuminated by monochromatic light with wavelength $\bar{\lambda}=589 \mathrm{~nm}, \boldsymbol{I}$ and 2 are identical tubes with air of length $\mathrm{l}^{-}=$ $=10.0 \mathrm{~cm}$ each, $D$ is a diaphragm with twe slits. After the air in tube $I$ was replaced with ammonia gas, the interference pattern on the screen $S c$ was displaced upwand by $N=17$ fringes. The refractive index of air is equal to $n=1.000277$. Determine the refractive index of ammonia gas.
5.78. An electromagnetic wave falls normally on the boundary between two isotropic dielectrics with refractive indices $n_{1}$ and $n_{4}$. Making use of the continuity condition for the tangential components, $\boldsymbol{E}$ and $\mathbf{H}$ across the boundary, demonstrate that at the interface the electric field vector $\mathbf{E}$
(a) of the transmitted wave experiences bo phase jump;
(b) of the reflected wave is subjected to the phase jump equal to $z$ if it is reflected from a medium of higher optical density.
5.79. A parallel beam of white light falls on a thin film whose refractive index is equal to $n=1.33$. The angle of indices is $\theta_{1}=$ $=52^{2}$. What must the film thickness be equal to for the reflected light to be coloured yellow $(\lambda=0.60 \mu \mathrm{m})$ most intensively?
5.80. Find the minimum thickness of a film with refractive index 1.33 at which light with wavelength $0.64 \mu \mathrm{m}$ experiences maximum reflection while light with wavelength $0.40 \mu \mathrm{m}$ is not reflected at all. The incidence angle of light is equal to $30^{\circ}$.
5.81. To decrease light losses due to reflection from the glass surface the latter is coated with a this layer of substance whose refractive index $n^{\prime}=\sqrt{n}$, where $n$ is the refractive index of the glass. In this case the amplitudes of electromagnetic oscillations reflected from both coated surfaces are equal. At what thickness of that ceating is the glass reflectivity in the direction of the normal equal to rero for light with wavelength 2?
5.82. Diffused monechrematie light with wavelength $\lambda=0.60 \mu \mathrm{m}$ falls on a thin film with refractive index $n=1.5$. Determine the film thickness if the angular separation of neighbouring maxima observed in reflected light at the angles close to $\theta=45^{\circ}$ to the nerma! is equal to $8 \theta^{\circ}=3.0^{\circ}$.
5.83. Monochromatie light passes threugh an orifice in a sereen $S c$ (Fig. 5.17) and being reflected from a thin transparent plate $P$ produces fringes of equal inclination on the sereen. The thickness of the plate is equal to $d$, the distance between the plate and the screen is $l$, the radil of the ith and kth dark rings are $r_{1}$ and $r_{2}$. Find the wavelength of light taking into account that $r_{i,}<i$.
5.84. A plane monochromatie light wave with wavelength $\lambda$ falls on the surface of a glass wedge whose faces form an angle $a<1$. The plane of incidence is perpendicular to the edge, the angle of incidence is $\theta_{1}$. Find the distance between the neighbouring fringe maxima on the sereen placed at right angles to reflected light.
5.85. Light with wavelength $\lambda=0.55 \mu \mathrm{m}$ from a distant point source falls normally on the surface of a glass wedge. Afringe pattern whose neighbouring maxima on the surface of the wedge are separated by a distance $\Delta x=0.21 \mathrm{~mm}$ is observed in reflected light. Find:
(a) the angle between the wedge faces;
(b) the degree of light monochromatism $(\Delta \lambda / \lambda)$ if the fringes disappear at a distance $t \simeq 1.5 \mathrm{~cm}$ from the wedge’s edge.
5.86. The convex surface of a plase-convex glass lens comes inte contact with a glass plate. The curvature radius of the lens’s convex surface is $\boldsymbol{R}$, the wavelength of light is equal to 2 . Find the width $\Delta r$ of a Newton ring as a function of its radius $r$ in the region where $\Delta r<r$.
5.87. The convex surface of a plano-convex glass lens with curvature radies $R=40 \mathrm{~cm}$ comes into contact with a glass plate. A certain ring ebserved in reflected light has a radius $r=2.5 \mathrm{~mm}$. Watching the given ring, the lens was gradually removed from the plate by a distance $\Delta h=5.0 \mu \mathrm{m}$. What has the radius of that ring become equal to?
5.88. At the crest of a spherical surlace of a plano-convex lens there is a ground-off plane spot of radius $r_{0}=3.0 \mathrm{~mm}$ through which the leas comes inte contact with a glass plate. The curvature radius of the lens’s convex surface is equal to $R=150 \mathrm{~cm}$. Find the radius of the sixth bright ring when observed in reflected light with wavelength $\lambda-655 \mathrm{~nm}$.
5.89. A plane-cenvex glass lens with curvature radius of spherical surface $R=12.5 \mathrm{~cm}$ is pressed agaisst a glass plate. The diameters of the tenth and fifteenth dark Newton’s ring in reflected light are equal to $d_{1}=1.00 \mathrm{~mm}$ and $d_{1}=1.50 \mathrm{~mm}$. Find the wavelength of light.
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5.90. Twe plane-convex thin glass lenses are brought inte contact with their spherical surfaces. Find the optical power of such a system If in reflected tight with wavelength $\lambda=0.60 \mu \mathrm{m}$ the diameter of the fifth bright ring is $d=1.50 \mathrm{~mm}$.
5.91. Two thin symmetric glass lenses, one biconvex and the other biconcave, are brought into contact to make a system with optical pewer $\Phi^{\prime}=0.50$ D. Newten’s rings are observed in reflected light with wavelength $\lambda=0.61 \mu \mathrm{m}$. Determine:
(a) the radius of the tenth dark ring:
(b) how the radius of that ring will change when the space between the lenses is filled up with water.
5.92. The spherical surface of a plane-convex lens comes inte contact with a glass plate. The space between the lens and the plate is filled up with carbon dioxide. The refractive indices of the lens, carbon dioxide, and the plate are equal to $n_{1}=1.50, n_{2}=1.63$, and $n_{3}=1.70$ respectively. The curvature radius of the spherical surface of the lens is equal to $R=100 \mathrm{~cm}$. Determine the radiss of the fifth dark Newton’s ring in reflected light with wavelength $\lambda=0.50 \mu \mathrm{m}$.
5.93. In a two-beam interferometer the orange mercury line composed of twe wavelengths $\lambda_{1}=576.97 \mathrm{~nm}$ and $\lambda_{1}=579.03 \mathrm{~nm}$ is employed. What is the least onder of interference at which the sharpness of the fringe pattern is the worst?
5.94. In Michelsen’s interferometer the yellow sodium line composed of two wavelengths $\lambda_{1}-589.0 \mathrm{~nm}$ and $\lambda_{2}-589.6 \mathrm{~nm}$ was used. In the process of translational displacement of one of the mirrors the interference pattern vanished periodically (why?). Find the displacement of the mirror between two successive appearances of the sharpest pattern.
5.95. When a Fabry-Perot étalon is illuminated by monechromatie light with wavelength $\lambda$ an interference pattern, the system of con-
Fie. S.18.
centric rings, appears in the focal plane of a lens (Fig. 5.18). The thickness of the étalon is equal to $d$. Determine how
(a) the position of rings;
(b) the angular width of fringes
depends on the order of interference.
5.96. For the Fabry-Perot étalon of thickness $d=2.5 \mathrm{~cm}$ find:
(a) the highest order of interference of light with wavelength $\lambda=0.50 \mu \mathrm{m}$
(b) the dispersion region $\Delta \lambda$, i.e. the spectral interval of wavelengths, within which there is still no overlap with other orders of interference if the observation is carried out approximately at wavelength $\lambda=0.50 \mu \mathrm{m}$.