(ii) State one advantage of a reflecting telescope over refracting telescope.

(i) Light diverging from a point source.

(ii) Light emerging out of a convex lens when a point source is placed at its focus.

(iii) The portion of a wave front of light from a distant star intercepted by the earth.

At what value of $$ x_{P} $$ do the rays have the maximum possible phase difference?

The positions $$ F, F^{\prime}, $$ and $$ H^{\prime} $$ (Fig. $$ c, $$ where the path of the ray of light is shown before and after passing through the system).

What is a wavefront ? How is it related to rays of light ? What is the shape of the wavefront at a point far away from the source of light ?

$$ \pi $$

$$(1)$$ Monochromatic light of smaller wavelength is used.

$$(2)$$ Slit is made narrower

From a far light source?

Light diverging from a point source.

How is the diffraction pattern it produces on a distant screen related to that of a vertical slit equal in width to the hair?

(a) whose thickness is equal to $$ d $$ and curvature radii of the surfaces are the same and equal to $$ R $$;

(a) In a single slit diffraction experiment, the width of the slit is made double the original width. How does this affect the size and intensity of the central diffraction band?

(b) In what way is diffraction from each slit related to the interference pattern in a double-slit experiment?

(c) When a tiny circular obstacle is placed in the path of light from a distant source, a bright spot is seen at the centre of the shadow of the obstacle. Explain why?

(d) Two students are separated by a 7 m partition wall in a room 10 m high. If both light and sound waves can bend around obstacles, how is it that the students are unable to see each other even though they can converse easily.

(e) Ray optics is based on the assumption that light travels in a straight line. Diffraction effects (observed when light propagates through small apertures/slits or around small obstacles) disprove this assumption. Yet the ray optics assumption is so commonly used in understanding location and several other properties of images in optical instruments. What is the justification?

Answer the following questions:

(a) When a low flying aircraft passes overhead, we sometimes notice a slight shaking of the picture on our TV screen. Suggest a possible explanation.

(b) As you have learnt in the text, the principle of linear superposition of wave displacement is basic to understanding intensity distributions in diffraction and interference patterns. What is the justification of this principle?

(b) Unpolarised light is incident on a Polaroid. How would the intensity of transmitted light change when the Polaroid is rotated?

i) a point source and ii) a line source?

a) Mention the phenomenon behind it.

b) Differentiate the interference pattern with a coherently illuminated single slit diffraction pattern.

(a) Determine the condition for maxima at the point $$P$$.

(b) Find the order of the centre bright fringe if $$d=0.5\ mm, D=1\ m$$

(All wavelength in this problem are for the given medium of refractive index 4/Ignore dispersion.) Find the location of central maxima (bring fringe with zero path difference ) on the y-axis.

i) Coming from a very far off source and

ii) Diverging radially from a point source

At what value of $$ x_{P} $$ do the rays have the minimum possible phase difference?

(i) Distinguish between unpolarised light and inearly polarised light. How does one get linearly light with the help of a Polaroid?

(ii) A narrow beam of unpolarised of light of intensity $$I_0$$ is incident on a Polaroid $$P_1$$. The light transmitted by it is then incident on a second Polaroid $$P_2$$ with its pass axis making angle of $$60^0$$ relative to the pass axis of $$P_1$$. FInd the intensity of the light transmitted by $$P_2$$.

(i) What is linearly polarized light? Describe briefly using a diagram how sunlight is polarised.

(ii) Unpolarised light is incident on a polaroid. How would the intensity of transmitted light change when the polaroid is rotated?

The vibrations in a beam of polarised light make an angle of $$60^0$$ with the axis of the polaroid sheet. What percentage of light is transmitted through the sheet?

From a point of source?

The portion of the wavefront of light from a distant star intercepted by the Earth.

Light emerging out of a convex lens when a point source is placed at its focus.

(b) Two wavelengths of sodium light 590 nm and 596 nm are used in turn to study the diffraction taking place at a single slit of aperture $$2 \times 10^{-6} m$$. The distance between the slit and the screen is 1.5 m. Calculate the separation between the positions of first maxima of the diffraction pattern obtained in the two cases.

(b) The ratio of the intensities at minima to the maxima in the Young's double slit experiment is 9:Find the ratio of the widths of the two slits.

When the width of the slit is made double the original width, how would this affect the size and intensity of the central diffraction band?

(b) In a single slit diffraction experiment, the width of the slit is made double the original width. How does this affect the size and intensity of the central diffraction bond? Explain.

(c) When a tiny circular obstacle is placed in the path of light from a distant source, a bright spot is seen at the centre of the obstacle. Explain why.

What multiple of $$ \lambda $$ gives the phase difference when $$ x_{P}=6.00 \lambda ?$$

Did this change in wavelength increase or decrease the diffraction of the signals into the shadow regions of obstacles?

(b) Derive a mathematical expression for the width of interference fringes obtained in Young's double slit experiment with the help of a suitable diagram.

(c) If s is the size of the source and b its distance from the plane of the two slits, what should be the criterion for the interference fringes to be seen?

OR

(a) In Young's double slit experiment, describe briefty how bright and dark fringes are obtained on the screen kept in front of a double slit. Hence obtain the expression for the fringe width.

(b) The ratio of the intensities at minima to the maxima in the Young's double slit experiment is 9 :Find the ratio of the widths of the two slits.

(a) Under that idealized assumption, estimate the smallest linear width on Earth's surface that the astronaut can resolve. Take the astronaut's pupil diameter to be $$ 5 \mathrm{mm} $$ and the wavelength of visible light to be $$ 550 \mathrm{nm} $$.

(b) Can the astronaut resolve the Great Wall of China (Fig. $$ 36-40 $$ ), which is more than $$ 3000 \mathrm{km} $$ long, $$5$$ to $$ 10 \mathrm{m} $$ thick at its base, $$ 4 \mathrm{m} $$ thick at its top, and $$ 8 \mathrm{m} $$ in height?

(c) Would the astronaut be able to resolve any unmistakable sign of intelligent life on Earth's surface?

One limitation on such a device is the spreading of the beam due to diffraction, with resulting dilution of beam intensity. Consider such a laser operating at a wavelength of $$ 1.40 \mathrm{nm} $$. The element that emits light is the end of a wire with diameter $$ 0.200 \mathrm{mm} $$

(a) Calculate the diameter of the central beam at a target $$ 2000 \mathrm{km} $$ away from the beam source.

(b) What is the ratio of the beam intensity at the target to that at the end of the wire? (The laser is fired from space, so neglect any atmospheric absorption.)

Let the reflecting planes have separation $$ d=0.252 \mathrm{nm} . $$

The crystal is turned through angle $$ \phi $$ around an axis perpendicular to the plane of the page until these reflecting planes give diffraction maxima.

What are the

(a) smaller and

(b) larger value of $$ \phi $$ if the crystal is turned clockwise and the

(c) smaller and

(d) larger value of $$ \phi $$ if it is turned counterclockwise?

Assuming that the lake width D is much greater than a and x, and that $$\lambda \geq a$$, find an expression that gives the values of x for which the signal at the receiver is maximum. (Hint : Does the reflection cause a phase change?)

Calculate their

(a) angular and

(b) linear separation, assuming ideal conditions. Take $$ \lambda=540 \mathrm{nm} $$ and the pupil diameter of the astronaut's eye to be $$ 5.0 \mathrm{mm} $$

How would the diffraction pattern of a single slit be affected when:

(i) The width of the slit is decreased?

(ii) The monochromatic source of light is replaced by a source of white light?

(i) Why are coherent sources necessary to produce a sustained interference pattern?

(ii) In Young's double slit experiment using monochromatic light of wavelength, the intensity of light at a point on the screen where path difference is, is K units. Find out the intensity of light at a point where path difference is.

(a) the angle $$\Delta \theta$$ between the direction to the Fraunhofer maximum of zeroth order and that of inceident light;

(b) angular width of the Fraunhofer maximum zeroth order.

(a) the image dimension $$y'$$ on the plate if the transverse dimension of the source is $$y = 6.0 mm$$ .

(b) the minimum height of irregularities, covering the surface of the ball at random, at which the ball obstructs light.

What is meant by coherent sources ? What are the two methods for obtaining coherent sources in the laboratory ?

(a) the angles $$\theta$$ at which the radiation intensity is maximum;

(b) the conditions under which the radiation intensity in the direction $$\theta = \pi$$ is maximum and in the opposite direction, minimum.

(a) in what order of spectrum these components will be resolved.

(b) the least difference of wavelengths that can be resolved by this grating in a wavelength region $$\lambda \approx 670 nm$$.

(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?

(a) the distance on the photographic plate between the components of this doublet in the spectra of the first and second order;

(b) whether these components will be resolved in these orders of the spectrum.

$$\bullet F S H H^{\prime} F^{\prime} $$

(b) whose refractive surfaces are concentric and have the curvature radii $$ R_{1} $$ and $$ R_{2}\left(R_{2}>R_{1}\right) $$

$$\dfrac{1}{\rho}=\dfrac{\partial}{\partial N}(\ln n)$$

where the derivative is taken with respect to the principal normal to the ray. Derive this formula, assuming 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 $$ \nabla n $$ at a given point.

(b) have the optical power equal to -1.0 D if the curvature radii of its convex and concave surfaces are equal to 10.0 and $$ 7.5 \mathrm{cm} $$ respectively?

(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.

Prove that the dispersion is given by

$$\dfrac{d \lambda}{d y}=\dfrac{L^{2} d}{m\left(L^{2}+y^{2}\right)^{3 / 2}}$$

(a) Let the thickness of the plate be $$ d . $$ Show that the phase difference between the $$ \mathrm{O} $$ ray and the $$ \mathrm{E} $$ ray after traveling the thickness of the plate is

$$\theta=\dfrac{2 \pi d}{\lambda}\left|n_{O}-n_{E}\right|$$

where $$ \lambda $$ is the wavelength in air.

(b) In a particular case, the incident light has a wavelength of 550 nm. Find the minimum value of $$ d $$ for a quartz plate for which $$ \theta=\pi / 2 . $$ Such a plate is called a quarter-wave plate. Use values of $$ n_{O} $$ and $$ n_{E} $$ from Table 38.1 .

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