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Showing posts from December, 2023

Analytical expression of intensity for constructive and destructive interference due to Young's double slit

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 $ $y= \left( a_{1} + a_{2} \: cos \phi \right) \: sin \omega t + a_{2} \: sin \phi \: cos \omega t \qquad(4)$

Viscosity, Viscous force and Coefficient of Viscosity

Definition of Viscosity: It is the property of a fluid that opposes the relative motion between its adjacent layers. This property of the fluid is known as viscosity. It is also called the resistance of fluid to flow or deformation or fluid thickness. Effect of temperature on Viscosity: The viscosity of the fluid decreases sharply with the temperature rise and becomes zero at boiling temperature. On the other hand, the viscosity of the gases increases with the temperature rise. Definition of Viscous Force (Internal Frictional Force): When a layer of fluid slide over another layer of the same fluid then an internal tangential frictional force act between them which opposes the relative motion between the layers. This tangential force is called viscous force or internal frictional force. In the absence of external force, the viscous force would soon bring the fluid to rest. Factor affecting the viscous force: There are the following factors that affect the v

Brief Description of Liquid Lasers

Brief Description: (Liquid Lasers) Due to their homogeneous properties and a very high optical cavity of liquids, these are also used as active materials in lasers. Liquid lasers are four-level lasers that use liquids as active material or lasing medium. In these lasers, laser tubes are filled with liquid instead of laser rods as in solid-state lasers or gas in gas lasers. Liquid laser medium has some advantages like very high gain, no cracking for high output power, feasibility of cooling the liquid by circulation, narrow frequency spectrum, etc. In liquid lasers, optical pumping is required for laser action. Optical pumping includes flash tubes, nitrogen lasers, excimer lasers, etc. A rare earth ion dissolved in a solution makes it possible to obtain optically pumped laser action in liquids. The first successful liquid laser was reported by using europium ions ($Eu^{+3}$) in which a sharp and strong laser transition was observed at $6131 A^{\circ}$ wavelength. In this laser, a eur

Principle of continuity in fluid

Statement of Principle of Continuity: When an ideal liquid (i.e. incompressible and non-viscous liquid ) flows in streamlined motion through a tube of non-uniform cross-section, then the product of the velocity of flow and area of cross-section is always constant at every point in the tube. Mathematical Analysis (Proof) Let us consider, an ideal liquid (i.e. incompressible and non-viscous liquid ) flow in streamline motion through a tube $XY$ of the non-uniform cross-section. Now Consider: The Area of cross-section $X = A_{1}$ The Area of cross-section $Y = A_{2}$ The velocity per second (i.e. equal to distance) of fluid at cross-section $X = v_{1}$ The velocity per second (i.e. equal to distance) of fluid at cross-section $Y = v_{2}$ The volume of liquid entering at the cross-section $X$ in $1$ second is = $A_{1}v_{1}$ The mass of liquid entering at the cross-section $X$ in $1$ second is = $ \rho A_{1}v_{1}$ Similarly, the mass of the liquid com

Combined Focal Length and Power of two thin lenses in contact

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

Refraction of light through a thin lens : Lens maker's formula

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

Refraction of light through the convex spherical surface

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(

Refraction of light through the concave spherical surface

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