Physics Vidyapith ✍️

Principle of Simple Microscope: The principle of the simple microscope is based on the magnification of an image by using a simple convex lens. Construction: A simple microscope consists of one convergent lens only. The object is placed between the lens and its focal length, and the eye is placed just behind the lens. Then the eye sees a magnified, erect, and virtual image on the same side as the object at the least distance of distinct vision $(D)$ from the eye, and the image is then seen most distinctly. Working: If the small object $ab$ is placed between a lens $O$ and its first focus $f$ then Its magnified virtual image $a_{1}b_{1}$ is formed at a distance $D$ from the lens. Since the eye is just behind the lens, the distance of image $a_{1}b_{1}$ from the eye is also $D$. Angular Magnification Or Magnifying Power($M$): The ratio of the angle subtended by the image at the eye ($\beta$) to the angle subtended by the object at the eye when placed at

The basic definition of Light: Light is a form of energy that produces the sensation of vision in the eye by which we can see objects. There are some facts about light as follows: 1. Lightwave moves along a straight line path. 2. Light waves can travel through vacuum and medium both. 3. Light is an electromagnetic wave. 4. A light wave is the transverse wave in nature. 5. Light can be dispersed. Besides these facts, light also shows the phenomenon of interference, diffraction, polarisation photoelectric effect, etc. To explain the above facts, many principles have been given from time to time, e.g., Newton's corpuscular theory, Huygen wave theory, Maxwell's principle of electromagnetic wave, Planck's quantum principle, dual nature of light, etc.

Principle: The principle of the compound microscope is based on the magnification of an image by using two lenses. Construction: A compound microscope consists of two convergent lenses (i.e. objective lens $O$ and eye-piece lens $e$) placed coaxially in a double tube system. The objective lens is an achromatic convergent lens system of short focal length and short aperture. The other eye-piece lens $e$ is also an achromatic convergent lens system of large focal length and large aperture. The observation is taken through the eye-piece lens by the observer. The eye-piece lens is fitted outer side of a movable tube and the inner side connects with a non-movable tube in which the objective lens is fitted on another side of the non-movable tube. The separation between the objective or eye-piece lens can be changed by an arrangement, this is known as rack and pinion arrangement. Working: Suppose a small object $ab$ is placed slightly away from the first focus $f_{\circ}$ of the obj

Prism Spectra 1.) Prism spectra are obtained by the phenomena of dispersion of light. 2.) Prism spectra have only one order. 3.) A prism spectrum is of bright intensity. 4.) In prism spectra, spectral colors overlap each other. 5.) Red color is dispersed least whereas violet color disperses the maximum. 6.) Prism spectrum depends upon the material of prism. 7.) The prism spectral lines are curved. . Grating Spectra 1.) Grating spectra are obtained by the phenomena of diffraction of light. 2.) Grating spectra has more than one order. 3.) Grating spectra is of less intensity. 4.) In grating spectra, there is no overlapping of color. 5.) Red color diffracts the maximum whereas violet color diffracts the least. 6.) Grating spectra are independent of the material of grating. 7.) The grating spectral lines are almost straight.

Interference 1.) It is due to the superposition of two or more than two wavefronts coming from coherent sources. 2.) The intensity of all bright fringes are same 3.) Interference fringes either of the same size or decrease after moving away from the center. 4.) Dark fringes are usually perfectly dark. 5.) A minimum coherent source is needed. Diffraction: 1.) It occurs due to secondary wavelets, originating from infinite different points of the same wavefronts. 2.) Central maxima of bright fringe is followed by either side maxima of decreasing intensity. 3.) Interference fringes are never of the same shape and size. 4.) Dark fringes are not perfectly dark. 5.) It is possible by either one or more than one source which need not be coherent.

Newton's Corpuscular Theory In the year 1675, Newton proposed the corpuscular theory of light to explain the existing phenomenon of light. There are the following assumptions of this theory: 1. The light consists of very small, lightweight, and invisible particles. These particles are known as corpuscles. 2. These corpuscles move with the velocity of light in a homogeneous medium in all possible directions in a straight line and they carry kinetic energy with them. 3. When these corpuscles fall on the retina of the eye, they produce the sensation of vision. 4. The size of corpuscles of different colors is different (ie, the color of light depends on the size of the corpuscle). (A) Success of Carpuscles Theory Based on this theory, the following facts related to light were explained successfully: 1. The light has energy: Since corpuscles have kinetic energy. Therefore, the energy of the light beam is due to the kinetic energy of the corpuscles. 2. Motion

Expression for fringe's width: Let us consider two wave from slit $S_{1}$ and $S_{2}$ superimpose on each other and form interfernece patteren on the screen. The distance between the two slits is $d$ and distance between slit to screen is $D$. Now take a $n^{th}$ fringe from the centre $O$ of the screen which is at distance $y_{n}$. So the path difference between the rays $\Delta x = S_{2}P- S_{1}P \quad(1)$ In $\Delta S_{1}PM$ $S_{1}P^{2}=S_{1}M^{2}+PM^{2} \quad(2)$ From figure: $S_{1}M =D$ $PM= y_{n}- \left(\frac{d}{2}\right) $ Now subtitute these values in equation $(2)$, then $S_{1}P^{2}=D^{2}+ \left( y_{n}- \frac{d}{2} \right)^{2} \quad(3)$ In $\Delta S_{2}PN$ $S_{2}P^{2}=S_{2}N^{2}+PN^{2} \quad(4)$ From figure: $S_{2}N =D$ $PN= y_{n} + \left(\frac{d}{2}\right) $ Now subtitute these values in equation $(4)$, then $S_{2}P^{2}=D^{2}+ \left( y_{n} + \frac{d}{2} \right)^{2} \quad(5)$ Now subtract the equation $(3)$ in equ

Analytical expression of intensity for interference due to Young's double slit: Let us consider two waves from slit $S_{1}$ and $S_{2}$ having amplitude $a_{1}$ and $a_{2}$ respectively superimpose on each other at point $P$ . If the displacement of waves is $y_{1}$ and $y_{2}$ and the phase difference is $\phi$ then $y_{1}=a_{1} \: sin \omega t \qquad(1)$ $y_{2}=a_{2} \: sin \left( \omega t + \phi \right) \qquad(2)$ According to the principle of superposition: $y=y_{1}+y_{2} \qquad(3)$ Now substitute the value of $y_{1}$ and $y_{2}$ in the above equation $(3)$ $y=a_{1} \: sin \omega t + a_{2} \: sin \left( \omega t + \phi \right)$ $y=a_{1} \: sin \omega t + a_{2} \left( sin \omega t \: cos \phi + cos \omega t \: sin \phi \right) $ $y=a_{1} \: sin \omega t + a_{2} \: sin \omega t \: cos \phi + a_{2}\: cos \omega t \: sin \phi $ $y= \left( a_{1} + a_{2} \: cos \phi \right) \: sin \omega t + a_{2} \: sin \phi \: cos \omega t \qquad(4)$

Derivation of the combined focal length and power of two thin lenses in contact: Case (1): When both are convex lens- a.) The combined focal length of two thin convex lenses in contact: Let us consider that two convex lenses $L_{1}$ and $L_{2}$ are connected with transparent cement Canada Balsam. If the focal length of the lenses is $f_{1}$ and $f_{1}$ and an object $O$ is placed at distance $u$ from the first lens $L_{1}$ and its image $I'$ is formed at a distance $v'$ from the first lens $L_{1}$. Therefore from the equation of focal length for lens $L_{1}$ $\frac{1}{f_{1}} = \frac{1}{v'} - \frac{1}{u} \qquad(1)$ For the second lens, The image $I'$ works as a virtual object for the second lens $L_{2}$ which image $I$ is formed at a distance $v$ from the second lens $L_{2}$. Therefore from the equation of focal length for lens $L_{2}$ $\frac{1}{f_{2}} = \frac{1}{v} - \frac{1}{v'} \qquad(2)$ Now add the equation $(1)$ and equation $(2)$. t

Derivation of refraction of light through a thin lens & Lens maker's formula: Let us consider, A convex lens having thickness $t$ and radius of curvature of surfaces is $R_{1}$ and $R_{2}$. If an object $O$ is placed at distance $u$ from the first surface of the convex lens and its image $I'$ is formed at distance $v'$ from the first surface of the convex lens then refraction of light through the first spherical surface of the lens $ \frac{\left( n_{2} - n_{1} \right)}{R_{1}} = \frac{n_{2}}{v'} - \frac{n_{1}}{u} \qquad(1) $ Now the Image $I'$ works as a virtual object for the second surface of the convex lens which image $I$ formed at distance $v$ from the second surface of the lens. So refraction of light through the second surface of the lens $ \frac{\left( n_{1} - n_{2} \right)}{R_{2}} = \frac{n_{1}}{v} - \frac{n_{2}}{v' - t} $ Here $t$ is the thickness of the lens. If the lens is very thin then thickness will be

Derivation of refraction of light through the convex spherical surface: Let us consider, a convex spherical surface which has radius of curvature $R$. If an object $O$ is placed at a distance $u$ from pole $P$ and its image $I$ is formed at distance $v$ from pole $P$ and the angle subtended by the object, image, and center of curvature is $\alpha$, $\beta$, and $\gamma$ then from figure In $\Delta MOC$ $i= \gamma + \alpha \qquad(1)$ In $\Delta MIC$ $r = \gamma + \beta \qquad(2)$ According to Snell's Law: $\frac{sin \: i}{sin \: r} = \frac{n_{2}}{n_{1}} \qquad(3)$ Here the aperture of the spherical surface is very small so point $M$ will be very close to point $P$ and angle $i$ and $r$ will be small. So $sin \: i \approx i$ $sin \: r \approx r$ So equation $(3)$ can be written as $\frac{ i}{ r} = \frac{n_{2}}{n_{1}} \qquad(4)$ Now subtitute the value of $i$ and $r$ from equation $(1)$ and equation $(2)$ in equation $(4)$ $\frac{ \left(

Derivation of refraction of light through the concave spherical surface: Let us consider, a concave spherical surface of radius of curvature $R$. If an object $O$ is placed at a distance $u$ from pole $P$ and its image $I$ is formed at distance $v$ from pole $P$ and the angle subtended by the object, image, and center of curvature is $\alpha$, $\beta$, and $\gamma$ then from figure In $\Delta MOC$ $\gamma= \alpha + i $ $i= \gamma - \alpha \qquad(1)$ In $\Delta MIC$ $\gamma= \beta + r $ $r = \gamma - \beta \qquad(2)$ According to Snell's Law: $\frac{sin \: i}{sin \: r} = \frac{n_{2}}{n_{1}} \qquad(3)$ Here the aperture of the spherical surface is very small so point $M$ will be very close to point $P$ and angle $i$ and $r$ will be small. So $sin \: i \approx i$ $sin \: r \approx r$ So equation $(3)$ can be written as $\frac{ i}{ r} = \frac{n_{2}}{n_{1}} \qquad(4)$ Now subtitute the value of $i$ and $r$ from equation $(1)$ and equatio

Definition of Refraction of light→ When a light goes from one medium to another medium then light bends from the path. This bending of the light phenomenon is knowns as the refraction of light . When light goes from a rarer medium to a denser medium, light bends toward the normal as shown in the figure below: Propagation of light from rarer medium to denser medium When light goes from a denser medium to a rarer medium, the light goes away from normal as shown in the figure below. Propagation of light from denser medium to rarer medium Properties of refraction of light→ The incident ray, normal ray, and refracted ray lie on the same point. According to Snell's law - The ratio of the sine of the incident angle to the sine of the refractive angle is always constant. This constant value is known as the refractive index of the medium. $\frac{sin \: i}{sin \: r}=constant(_{1}n_{2}) $ $\frac{sin \: i}{sin \: r}= (_{1}n_{2})$

Difference between Fraunhofer Diffraction and Fresnel Diffraction→ S.No. Fresnel Diffraction Fraunhofer Diffraction 1. The distance between source to slit and slit to screen is finite. The distance between source to slit and slit to screen is infinite. 2. The shape of the incident wavefront on the slit is spherical or cylindrical. The shape of the incident wavefront on the slit is plane. 3. The shape of the incident wavefront on the screen is spherical or cylindrical. The shape of the incident wavefront on the screen is a plane. 4. There is a path difference created between the rays before entering the slit. This path difference depends on the distance between the source and slit. There is not any path difference between the rays before entering the slit. 5. Path difference between the rays forming the diffraction pattern depends on the

The equation for missing order in the double-slit diffraction pattern→ The nature of the diffraction pattern due to the double slits depends upon the relative values of $e$ and $d$. If, however, $e$ is kept constant and $d$ is varied, then certain orders of interference maxima will be missing. We know that, the direction of interference maxima $(e+d)\:sin\theta=\pm n\lambda \qquad(1)$ The direction of diffraction minima $e \: sin\theta=\pm m\lambda \qquad(2)$ Divide the equation $(1)$ by equation $(2)$ $\frac{(e+d)}{e}=\frac{n}{m}$ Case (I)→ If $e=d$ then n=2m So for $m=1,2,3,....$ The $n=2,4,6,....$ Thus, the $2_{nd}, 4^{th}, 6^{th}, ...$ order interference maxima will be missing. Case (II) → If $e=\frac{d}{2}$ then n=3m So for $m=1,2,3,....$ The $n=3,6,9,....$ Thus, the $3_{rd}, 6^{th}, 9^{th}, ...$ order interference maxima will be missing.

A diffraction grating (or $N$-slits) consists of a large number of parallel slits of equal width and separated from each other by equal opaque spaces. It may be constructed by ruling a large number of parallel and equidistance lines on a plane glass plate with the help of a diamond point. the duplicates of the original grating are prepared by pouring a thin layer of colloidal solution over it and then allowed to Harden. This layer is then removed from the original grating and fixed between two glass plates which serve as a plane transmission grating. Generally, A plane transmission grating has 10000 to 15000 lines per inch. Diffraction due to N- slits OR Grating Theory→ Since plane diffraction grating is an $N$-slit arrangement, the deflection pattern due to it will be the combined diffraction effect of all such slits. Let a plane wavefront of monochromatic light be incident normally on the $N$-parallel slit of the gratings. Each point within the slits then sends out s

Let a plane wavefront be incident normally on slit $S_{1}$ and $S_{2}$ of equal $e$ and separated by an opaque distance $d$.The diffracted light is focused on the screen $XY$. The diffracted pattern on the screen consists of equally spaced bright and dark fringe due to interference of light from both the slits and modulated by diffraction pattern from individual slits. op The diffraction pattern due to double-slit can be explained considering the following points → All the points in slits $S_{1}$ and $S_{2}$ will send secondary waves in all directions. All the secondary waves moving along the incident wave will be focussed at $P$ and the diffracted waves will be focussed at $P'$ The amplitude at $P'$ is the resultant from two slit each of amplitude $R=\frac{A\:sin\alpha}{\alpha}$ T two waves from two-slit $S_{1}$ and $S_{2}$ will interfere at $P'$ Fraunhofer diffraction due to double slits Expression for Intensity → $\Delta = S_{2}M$

Dispersive power of plane diffraction grating: The dispersive power of a diffraction grating is defined as: The rate of change of the angle of diffraction with the change in the wavelength of light are called dispersive power of plane grating. If the wavelenght changes from $\lambda$ to $\lambda +d\lambda$ and respective change in the angle of diffraction be from $\theta$ to $\theta+d\theta$ then the ratio $\left(\frac{d\theta}{d\lambda} \right)$ Expression of Dispersive power of a plane diffraction grating: The grating equation for a plane transmission grating for normal incidence is given by $(e+d)sin\theta=n\lambda \qquad(1)$ Where$(e+d)$ - Grating Element$\qquad \:\: \theta$ - Diffraction angle for spectrum of $n^{th}$ order Differentiating equation $(1)$ with respect to $\lambda$, we have $(e+d)cos\theta \left( \frac{d\theta}{d\lambda} \right)=n$ $\frac{d\theta}{d\lambda}=\frac{n}{(e+d)cos\theta}$ $\frac{d\theta}{d\lambda}=\frac{n}{(e+d)\sqrt{

Let $S$ be a point monochromatic source of light of wavelength $\lambda$ placed at the focus of collimating lens $L_{1}$. The light beam is incident normally from $S$ on a narrow slit $AB$ of width $e$ and is diffracted from it. The diffracted beam is focused at the screen $XY$ by another converging lens $L_{2}$. The diffraction pattern having a central bright band followed by an alternative dark and bright band of decreasing intensity on both sides is obtained. Analytical Explanation: The light from the source $S$ is incident as a plane wavefront on the slit $AB$. According to Huygens's wave theory, every point in $AB$ sends out secondary waves in all directions. The undeviated ray from $AB$ is focused at $C$ on the screen by the lens $L_{2}$ while the rays diffracted through an angle $\theta$ are focussed at point $p$ on the screen. The rays from the ends $A$ and $B$ reach $C$ in the same phase and hence the intensity is maximum. Fraunhofer diffraction due to