– Spectral response of an eye $V(\lambda)$ is shown in Fig. 5.1.
Fig. 5.1.
– Luminous intensity $I$ and illuminance $E$ :
\[
I=\frac{d \Phi}{d \Omega}, \quad E=\frac{d \Phi_{i n c}}{d S} .
\]
– Illuminance produced by a point isotropic source:
\[
E=\frac{l \cos \alpha}{r^{2}},
\]
where $\alpha$ is the angle between the normal to the surface and the direction to the source.
– Luminosity $M$ and luminance $L$ :
\[
M=\frac{d \Phi_{\text {emit }}}{d S}, L=\frac{d \Phi}{d \Omega \Delta S \cos \theta} .
\]
– For a Lambert source $L=$ const and luminosity
\[
M=\pi L .
\]
– Relation between refractive angle $\theta$ of a prism and least deviation angle $\alpha$ :
\[
\sin \frac{\alpha+\theta}{2}=n \sin \frac{\theta}{2},
\]
where $n$ is the refractive index of the prism.
– Equation of spherical mirror:
\[
\frac{1}{s^{\prime}}+\frac{1}{s}=\frac{2}{R},
\]
where $R$ is the curvature radius of the mirror.
– Equations for aligned optical system (Fig. 5.2):
\[
\frac{n^{\prime}}{s^{\prime}}-\frac{n}{s}=\Phi, \frac{f^{\prime}}{s^{\prime}}+\frac{f}{s}=1, \quad x x^{\prime}=f f^{\prime} .
\]

Fig. 5.2.
– Relations between focal lengths and optical power:
\[
f^{\prime}=\frac{n^{\prime}}{\Phi}, \quad f=-\frac{n}{\Phi}, \quad \frac{f^{\prime}}{f}=-\frac{n^{\prime}}{n} .
\]
– Optical power of a spherical refractive surface:
\[
\Phi=\frac{n^{\prime}-n}{R} .
\]
– Optical power of a thin lens in a medium with refractive index $n_{0}$ :
\[
\Phi=\left(n-n_{0}\right)\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right),
\]
where $n$ is the refractive index of the lens.
– Optical power of a thick lens:
\[
\Phi=\Phi_{1}+\Phi_{2}-\frac{d}{n} \Phi_{1} \Phi_{2},
\]
where $d$ is the thickness of the lens. This equation is also valid for a system of two thin lenses separated by a medium with refractive index $n$.
200
– Principal planes $H$ and $H^{\prime}$ are removed from the crest points $O$ and $O^{\prime}$ of surfaces of a thick lens (Fig. 5.3) by the following distances:
\[
X=\frac{d}{n} \frac{\Phi_{2}}{\Phi}, X^{\prime}=-\frac{d}{n} \frac{\Phi_{1}}{\Phi} .
\]

Fig. 5.3.
– Lagrange-Helmholtz invariant:
\[
\text { ny } u=\text { const. }
\]
– Magnifying power of an optical device:
\[
\Gamma=\frac{\tan \psi^{\prime}}{\tan \psi},
\]
where $\psi^{\prime}$ and $\psi$ are the angles subtended at the eye by an image formed by the optical device and by the corresponding object at a distance for convenient viewing (in the case of a microscope or magnifying glass that distance is equal to $\left.l_{0}=25 \mathrm{~cm}\right)$.
5.1. Making use of the spectral response curve for an eye (see Fig. 5.1), find:
(a) the energy flux corresponding to the luminous flux of $1.0 \mathrm{~lm}$ at the wavelengths 0.51 and $0.64 \mu \mathrm{m}$;
(b) the luminous flux corresponding to the wavelength interval from 0.58 to $0.63 \mu \mathrm{m}$ if the respective energy flux, equal to $\Phi_{e}=$ $=4.5 \mathrm{~mW}$, is uniformly distributed over all wavelengths of the interval. The function $V(\lambda)$ is assumed to be linear in the given spectral interval.
5.2. A point isotropic source emits a luminous flux $\Phi=10 \mathrm{~lm}$ with wavelength $\lambda=0.59 \mu \mathrm{m}$. Find the peak strength values of electric and magnetic fields in the luminous flux at a distance $r=$ $=1.0 \mathrm{~m}$ from the source. Make use of the curve illustrated in Fig. 5.1.
5.3. Find the mean illuminance of the irradiated part of an opaque sphere receiving
(a) a parallel luminous flux resulting in illuminance $E_{0}$ at the point of normal incidence;
(b) light from a point isotropic source located at a distance $l=$ $=100 \mathrm{~cm}$ from the centre of the sphere; the radius of the sphere is $R=60 \mathrm{~cm}$ and the luminous intensity is $I=36 \mathrm{~cd}$.
5.4. Determine the luminesity of a surface whese luminance depends on direction as $L=L, \cos \theta$, where $\theta$ is the angle between the radlation direction and the normal to the surface.
5.5. A certain luminous surface obeys Lambert’s law. Its luminance is equal to $L$. Find:
(a) the luminous fux emitted by an element $\Delta S$ of this surface inte a cone whose axis is normal to the given element and whose aperture angle is equal to $\theta$;
(b) the luminosity of such a source.
5.6. An illuminant shaped as a plane horizental dise $S=100 \mathrm{~cm}^{2}$ in area is suspended ever the centre of a round table of radius $R=$ $-1.0 \mathrm{~m}$. Its luminance does net depend on direction and is equal to $L=1.6 \cdot 10^{4} \mathrm{ed} / \mathrm{m}^{2}$. At what height over the table should the illuminant be suspended to provide maximum illuminance at the circumference of the table? How great will that illuminance be? The illuminant is assumed to be a point source.
5.7. A point source is suspended at a height $h=1.0 \mathrm{~m}$ over the centre of a round table of radius $R=1.0 \mathrm{~m}$. The luminous intensity $I$ of the source depends on direction so that illuminance at all points of the table is the same. Find the function $I(\theta)$, where $\theta$ is the angle between the radiation direction and the vertical, as well as the luminous flux reaching the table if $I(0)=I_{0}=100 \mathrm{~cd}$.
5.8. A vertical shaft of light from a projecter forms a light spot $S=100 \mathrm{~cm}^{2}$ in area on the ceiling of a round roem of radius $R$ $-2.0 \mathrm{~m}$. The illuminance of the spot is equal to $E=1000 \mathrm{~lx}$. The reflection coefficient of the ceiling is equal to $p=0.80$. Find the maximum illuminance of the wall produced by the light reflected from the ceiling. The reflection is assumed to ebey Lambert’s law.
5.9. A luminous dome shaped as a hemisphere rests on a horizontal plane. Its luminosity is uniform. Determine the illuminance at the centre of that plane if its luminance equals $L$ and is independent of direction.
5.10. A Lambert source has the torm of an infinite plane. Its Iuminance is equal to $\boldsymbol{L}$. Find the illuminance of an area element eriented parallel to the given source.
5.11. An illuminant shaped as a plane herisontal dise of radius $\boldsymbol{R}=25 \mathrm{~cm}$ is suspended over a table at a heizht $h=75 \mathrm{~cm}$. The illuminance of the table below the centre of the illuminant is equal to $E_{0}=70 \mathrm{~lx}$. Assuming the source to obey Lambert’s law, find its Iuminesity.
5.12. A small lamp having the form of a uniformly luminous sphere of radius $\boldsymbol{R}-6.0 \mathrm{~cm}$ is suspended at a height $h=3.0 \mathrm{~m}$ above the floor. The luminance of the lamp is equal to $L=2.0 \cdot 10^{4} \mathrm{~cd} / \mathrm{m}^{\prime}$ and is independent of direction. Find the illuminance of the floor directly below the lamp.
5.13. Write the law of reflection of a light beam from a mirror in vector form, using the directing unit vectors $e^{\prime}$ and $e^{\prime}$ of the inci-
va
dent and reflected beams and the unit vector $\mathbf{n}$ of the outside normal to the mirror surface.
5.14. Demenstrate that a light beam reflected from three mutually perpendicular plane mirrors in succession reverses its direction.
5.15. At what value of the angle of incident $\theta_{1}$ is a shaft of light reflected from the surface of water perpendicular to the refracted shaft?
5.16. Two optical media have a plane boundary between them. Suppose $\theta_{10}$ is the critical angle of incidence of a beam and $\theta_{2}$ is the angle of incidence at which the refracted beam is perpendicular to the reflected one (the beam is assumed to come from an optically denser medium). Find the relative refractive index of these media if $\sin \theta_{1 e r} / \sin \theta_{1}=\eta=1.28$.
5.17. A light beam falls upon a plane-parallel glass plate $d=6.0 \mathrm{~cm}$ in thickness. The angle of incidence is $\theta=60^{\circ}$. Find the value of deflection of the beam which passed through that plate.
5.18. A man standing on the edge of a swimming pool looks at a stone lying on the bottom. The depth of the swimming pool is equal to $h$. At what distance from the surface of water is the image of the stone formed if the line of vision makes an angle $\theta$ with the normal to the surface?
5.19. Demenstrate that in a prism with small refracting angle $\theta$ the shaft of light deviates through the angle $a \simeq(n-1) \theta$ regardless of the angle of incidence, previded that the latter is also small.
5.20. A shaft of light passes through a prism with refracting angle $\theta$ and refractive index $n$. Let $a$ be the diffraction angle of the shaft. Demenstrate that if the shaft of light passes threugh the prism symmetrically,
(a) the angle $a$ is the least;
(b) the relationship between the angles $\alpha$ and $\theta$ is defined by Bq. (5.1e).
5.21. The least deflection angle of a certain glass prism is equal to its refracting angle. Find the latter.
5.22. Find the minimum and maximum deflection angles for a light ray passing through a glass prism with refracting angle $\theta=60^{\circ}$.
5.23. A trihedral prism with refracting angle $60^{\circ}$ provides the least deflection angle $37^{\circ}$ in air. Find the least deflection angle of that prism in water.
5.24. A light ray composed of two monochromatic components passes through a trihedral prism with refracting angle $\theta=60^{\circ}$. Find the angle $\Delta a$ between the components of the ray after its pass: age through the prism if their respective indices of refraction are equal to 1.515 and 1.520 . The prism is oriented to provide the least deflection angle.
5.2. Using Fermat’s principle derive the laws of deflection and refraction of light on the plane interface between two media.
5.26. By means of plotting find:
(a) the path of a light ray after reflection from a concave and convex spherical mirrors (see Fig. 5.4, where $F$ is the focal point, $O O^{\prime}$ is the optical axis);
Fig. 5.4.
(b) the positions of the mirror and its focal point in the cases illustrated in Fig. 5.5, where $P$ and $P^{\prime}$ are the conjugate points.
Fig. 5.5.
5.27. Determine the focal length of a concave mirror if:
(a) with the distance between an object and its image being equal to $l=15 \mathrm{~cm}$, the transverse magnification $\beta=-2.0$;
(b) in a certain position of the object the transverse magnification is $\beta_{1}=-0.50$ and in another position displaced with respect to the former by a distance $l=5.0 \mathrm{~cm}$ the transverse magnification $\beta_{2}=$ $=-0.25$.
5.28. A point source with luminous intensity $I_{0}=100 \mathrm{~cd}$ is positioned at a distance $s=20.0 \mathrm{~cm}$ from the crest of a concave mirror with focal length $f=25.0 \mathrm{~cm}$. Find the luminous intensity of the reflected ray if the reflection coefficient of the mirror is $\rho=0.80$.
5.29. Proceeding from Fermat’s principle derive the refraction formula for paraxial rays on a spherical boundary surface of radius $R$ between media with refractive indices $n$ and $n^{\prime}$.
5.30. A parallel beam of light falls from vacuum on a surface enclosing a medium with refractive index $n$ (Fig. 5.6). Find the shape of that surface, $x(r)$, if the beam is brought into focus at the point $F$ at a distance $f$ from the crest $O$. What is the maximum radius of a beam that can still be focussed?
204
5.31. A point source is located at a distance of $20 \mathrm{~cm}$ from the front surface of a symmetrical glass biconvex lens. The lens is $5.0 \mathrm{~cm}$ thick and the curvature radius of its surfaces is $5.0 \mathrm{~cm}$. How far beyond the rear surface of this lens is the image of the source formed?
5.32. An object is placed in front of convex surface of a glass plano-convex lens of thickness $d=9.0 \mathrm{~cm}$. The image of that object is formed on the plane surface of the lens serving as a screen. Find:
(a) the transverse magnification if the curvature radius of the lens’s convex surface is $R=2.5 \mathrm{~cm}$;
(b) the image illuminance if the luminance of the object is $L=$ $=7700 \mathrm{~cd} / \mathrm{m}^{2}$ and the entrance aperture diameter of the lens is $D=5.0 \mathrm{~mm}$; losses of light are negligible.
5.33. Find the optical power and the focal lengths
(a) of a thin glass lens in liquid with refractive index $n_{0}=1.7$ if its optical power in air is $\Phi_{0}=-5.0 \mathrm{D}$;
(b) of a thin symmetrical biconvex glass lens, with air on one side and water on the other side, if the optical power of that lens in air is $\Phi_{0}=+10 \mathrm{D}$.
5.34. By means of plotting find:
(a) the path of a ray of light beyond thin converging and diverging lenses (Fig. 5.7, where $O O^{\prime}$ is the optical axis, $F$ and $F^{\prime}$ are the front and rear focal points);
Fig. 5.7.
(b) the position of a thin lens and its focal points if the position of the optical axis $O O^{\prime}$ and the positions of the cojugate points $P, P^{\prime}$ (see Fig. 5.5) are known; the media on both sides of the lenses are identical;
(c) the path of ray 2 beyond the converging and diverging lenses (Fig. 5.8) if the path of ray 1 and the positions of the lens and of its
Fig. 5.8.
optical axis $O O^{\prime}$ are all known; the media on both sides of the lenses are identical.
5.35. A thin converging lens with focal length $f=25 \mathrm{~cm}$ projects the image of an object on a screen removed from the lens by a dis-
tance $l=5.0 \mathrm{~m}$. Then the screen was drawn closer to the lens by a distance $\Delta l=18 \mathrm{~cm}$. By what distance should the object be shifted for its image to become sharp again?
5.36. A source of light is located at a distance $l=90 \mathrm{~cm}$ from a screen. A thin converging lens provides the sharp image of the source when placed between the source of light and the screen at two positions. Determine the focal length of the lens if
(a) the distance between the two positions of the lens is $\Delta l=$ – $30 \mathrm{can}$;
(b) the transverse dimensions of the image at one position of the lens are $\eta=4.0$ greater than those at the other position.
5.37. A thin converging lens is placed between an ebject and a screen whose positions are fixed. There are two positions of the lens at which the sharp image of the object is formed on the screen. Find the transverse dimension of the object if at ene position of the lens the image dimension equals $h^{\prime}=2.0 \mathrm{~mm}$ and at the other, $h^{*}=4.5 \mathrm{~mm}$.
5.38. A thin converging lens with aperture ratio $D: f=1: 3.5$ ( $D$ is the lens diameter, $f$ is its local length) provides the image of a sufficiently distant object on a photographic plate. The object luminance is $L=260 \mathrm{~cd} / \mathrm{m}^{2}$. The losses of light in the lens amount to $a=0,10$. Find the illuminance of the image.
5.39. How does the luminance of a real image depend on diameter $D$ of a thin converging lens if that image is observed
(a) directly;
(b) on a white screen backscattering according to Lambert’s law?
5.40. There are two thin symmetrical lenses: one is converging. with refractive index $n_{1}=1.70$, and the other is diverging with refractive index $n_{3}=1.51$. Both lenses have the same curvature radius of their surfaces equal to $R=10 \mathrm{~cm}$. The lenses were put close tegether and submerged into water. What is the focal length of this system in water?
5.41. Determine the focal length of a concave spherical mirror which is manufactured in the form of a thin symmetric biconvex glass lens one of whose surfaces is silvered. The curvature radius of the lens surface is $\boldsymbol{R}=40 \mathrm{~cm}$.
5.42. Figure 5.9 illustrates an aligned system consisting of three thin lenses. The system is located in air. Determine:
กe. S.9.
(a) the position of the point of convergence of a parallel ray incoming from the left after passing through the system;
(b) the distance between the first fens and a point lying on the axis to the left of the system, at which that point and its image are located symmetrically with respect to the lens system.
5.43. A Galilean telescope of 10-fold magnification has the length of $45 \mathrm{~cm}$ when adjusted to infisity. Determine:
(a) the focal lengths of the telescope’s objective and ocular;
(b) by what distance the ocular should be displaced to adjust the telescope to the distance of $50 \mathrm{~m}$.
5.44. Find the magnification of a Keplerian telescope adjusted to infinity if the mounting of the objective has a diameter $D$ and the image of that mounting formed by the telescope’s ocular has a diameter $d$.
5.45. On passing through a telescope a flux of light increases its intensity $7=4.0 \cdot 10^{4}$ times. Find the angular dimension of a distant object if its image formed by that telescope has an angular dimension $y^{\prime}=2.0^{\circ}$.
5.46. A Keplerian telescope with magnification $\mathrm{r}-15$ was submerged into water which filled up the inside of the telescope. To make the system work as a telescope again within the former dimensions, the objective was replaced. What has the magnification of the telescopc become equal to? The refractive index of the glass of which the ecular is made is equal to $n=1.50$.
5.47. At what magnification $r$ of a telescope with a diameter of the objective $D=6.0 \mathrm{~cm}$ is the illuminance of the image of an object on the retina not less than without the telescope? The pupil diameter is assumed to be equal to $d_{0}-3.0 \mathrm{~mm}$. The losses of light in the telescope are negligible.
5.48 . The optical powers of the ebjective and the ecular of a microscope are equal to 100 and $20 \mathrm{D}$ respectively. The microscope magnification is equal to 50 . What will the magnification of the microscope be when the distance between the ebjective and the ocular is increased by $2.0 \mathrm{~cm}$ ?
5.49. A microsepe has a sumerical aperture sin $\alpha-0.12$, where $\alpha$ is the aperture angle subtended by the entrance pupil of the microscope. Assuming the diameter of an eye’s pupil to be equal to $d_{o}=$ – $4.0 \mathrm{~mm}$, determine the microscope magnification at which
(a) the diameter of the beam of light coming from the microscope is equal to the diameter of the eye’s pupil;
(b) the illuminance of the image on the retina is independent of magnification (consider the case when the beam of light pasing through the system \”microscope-eye\” is bounded by the mounting of the objective).
5.50. Find the positions of the prineipal planes, the focal and nodal points of a this biconvex symmetric glass lens with curvature radius of its surfaces equal to $\boldsymbol{H}=7.50 \mathrm{~cm}$. There is air on one side of the lens and water on the other.
5.51. By means of plotting find the positions of focal points and principal planes of aligned optical systems illustrated in Fig. 5.10:
(a) a telephoto lens, that is a combination of a converging and a diverging thin lenses $\left(f_{1}=1.5 \mathrm{a}, f_{2}=-1.5 \mathrm{a}\right)$;
Fig. S. 10
(b) a system of two thin converging lenses $\left(f_{1}=1.5 \mathrm{a}, f_{1}=0.5 \mathrm{a}\right.$ );
(c) a thick convex-concave lens $\left(d=4 \mathrm{~cm}, n=1.5, \Phi_{1}=+50 \mathrm{D}\right.$; $\left.\Phi_{1}=-50 \mathrm{D}\right)$.
5.52. An optical system is located in air. Let $O O^{\prime}$ be its optical axis, $F$ and $F^{\prime}$ are the front and rear focal points, $H$ and $H^{\prime}$ are the front and rear principal planes, $P$ and $P^{\prime}$ are the conjugate points. By means of plotting find:
(a) the positions $F^{\prime}$ and $H^{\prime}$ (Fig. 5.11a);
(b) the position of the point ‘ $s$ ‘ conjugate to the point $s$ (Fig. 5.116);
Fig. S,1t.
(c) the positions $F, F^{\prime}$, and $H^{\prime}$ (Fig. $5.11 \mathrm{c}$, where the path of the ray of light is shown before and after passing through the system). 5.53. Suppose $F$ and $F^{\prime}$ are the frent and rear focal points of an eptical system, and $\boldsymbol{H}$ and $\boldsymbol{H}^{\prime}$ are its front and rear principal points. By means of plotting find the position of the image $S^{\prime}$ of the point $S$ for the following relative positions of the points $\boldsymbol{S}, \boldsymbol{F}, \boldsymbol{F}^{\prime}, \boldsymbol{H}$, and $\boldsymbol{H}^{\prime}$ :
(a) PSHH’F; (b) $H S F^{\prime} F H^{\prime}$; (e) $H^{\prime} S F^{\prime} F H$; (d) $F^{\prime} H^{\prime} S H F$.
5.54. A telephoto lens consists of two thin lenses, the front converging lens and the rear diverging lens with eptical powers $\Phi_{1}=$ $=+10 \mathrm{D}$ and $\Phi_{1}=-10 \mathrm{D}$. Find:
(a) the focal length and the positions of principal axes of that system if the lenses are separated by a distance $d=4.0 \mathrm{~cm}$;
(b) the distance $d$ between the lenses at which the ratio of a focal length $f$ of the system to a distance $l$ between the converging lens and the rear principal focal point is the highest. What is this ratio equal te?
5.55. Calculate the positions of the principal planes and focal points of a thick convex-concave glass lens if the curvature radius of the convex surface is equal to $R_{1}=10.0 \mathrm{~cm}$ and of the concave surface to $R_{1}=5.0 \mathrm{~cm}$ and the lens thickness is $d=3.0 \mathrm{~cm}$.
5.56. An aligned eptical system consists of two thin lenses with focal lengths $f_{1}$ and $f_{2}$, the distance between the lenses being equal to d. The given system has to be replaced by one thin lens which, at any position of an object, would provide the same transverse magnification as the system. What must the focal length of this lens be equal to and in what position must it be placed with respect to the two-lens system?
5.57. A system consists of a thin symmetrical converging glass lens with the curvature radius of its surfaces $\boldsymbol{R}=38 \mathrm{~cm}$ and a plane mirrer oriented at right angles to the optical axis of the lens. The distance between the lens and the mirror is $l=12 \mathrm{~cm}$. What is the eptical power of this system when the space between the lens and the mirror is flled up with water?
5.58. At what thickness will a thick convex-concave glass lens in air
(a) serve as a telescope provided the curvature radius of its convex surface is $\Delta R=1.5 \mathrm{~cm}$ greater than that of its concave surface?
(b) have the epticai power equal to -1.0 D if the curvature radif of its cenvex and concave surfaces are equal to 10.0 and $7.5 \mathrm{~cm}$ respectively?
5.59. Find the positions of the principal plases, the focal length and the sign of the optical power of a thick convex-concave glass lens.
(a) whose thickness is equal to $d$ and curvature radii of the surfaces are the same and equal to $\boldsymbol{B}$;
(b) whese refractive surfaces are concentric and have the curvature radii $\boldsymbol{R}_{1}$ and $\boldsymbol{R}_{3}\left(\boldsymbol{R}_{2}>\boldsymbol{R}_{1}\right)$.
5.60. A telescope system consists of two glass balls with radii $R_{1}=5.0 \mathrm{~cm}$ and $R_{2}=1.0 \mathrm{~cm}$. What are the distance between the centres of the balls and the magnification of the system if the bigger ball serves as an ebjective?
5.61. Two identical thick symmetrical biconvex lenses are put elose together. The thickness of each lens equals the curvature radius of its surfaces, i.e. $d=R=3.0 \mathrm{~cm}$. Find the eptical power of this system in air.
5.62. A ray of light propagating in an isotropic medium with refractive index $n$ varying gradually from point to point has a curvature radius $\rho$ determined by the fermula
\[
\frac{1}{p}=\frac{\partial}{\partial N}(\ln n) \text {. }
\]
where the derivative is taken with respect to the prineipal normal to the ray. Derive this formuls, ssouming that in such a medium the law of refraction $n \sin \theta=$ const holds. Here $\theta$ is the angle between the ray and the direction of the vector $
abla_{n}$ at a given point.
5.63. Find the curvature radius of a ray of light propagating in a horizontal direction close to the Barth’s surface where the gradient of the refractive index in air is equal to approximately 3. $10^{-} \mathrm{m}^{-1}$. At what value of that gradient would the ray of light propagate all the way round the Earth?