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Work energy theorem Statement and Derivation

Work-energy theorem statement: The work between the two positions is always equal to the change in kinetic energy between these positions. This is known as the work energy Theorem. $W=K_{f}-K_{i}$ $W=\Delta K$ Derivation of the Work-energy theorem: According to the equation of motion: $v^{2}_{B}=v^{2}_{A}-2as $ $2as=v^{2}_{B}-v^{2}_{A}$ $2mas=m(v^{2}_{B}-v^{2}_{A})$ $mas=\frac{m}{2} (v^{2}_{B}-v^{2}_{A})$ $Fs=\frac{1}{2}mv^{2}_{B}-\frac{1}{2}mv^{2}_{A} \qquad (\because F=ma)$ $W=\frac{1}{2}mv^{2}_{B}-\frac{1}{2}mv^{2}_{A} \qquad (\because W=Fs)$ $W=K_{f}-K_{i}$ Where $K_{f}$= Final Kinetic Energy at position $B$ $K_{i}$= Initial Kinetic Energy at position $A$ $W=\Delta K$ Alternative Method (Integration Method): We know that the work done by force on a particle from position $A$ to position $B$ is- $W=\int F ds$ $W=\int (ma)ds \qquad (\

Davisson and Germer's Experiment and Verification of the de-Broglie Relation

Davisson and Germer's Experiment on Electron Diffraction: Davisson and Germer's experiment verifies the wave nature of electrons with the help of diffraction of the electron beam as wave nature exhibits the diffraction phenomenon. Principle: The principle of Davisson and Germer's experiment is based on the diffraction phenomenon of the electron beam by crystal and it verifies the de-Broglie relation. Theoretical Formula: If a narrow beam of electrons is accelerated by a potential difference $V$ volts, the kinetic energy $K$ acquired by each electron in the beam is given by $K=eV \qquad(1)$ Where $e$ is the charge of an electron The de-Broglie wavelength is given by $\lambda = \frac{h}{\sqrt {2m_{\circ} K \left( 1+ \frac{E_{K}}{2m_{\circ}c^{2}} \right)}}$ If $E_{K} \lt \lt 2m_{\circ}c^{2}$, then the term $\frac{E_{K}}{2m_{\circ}c^{2}}$ will be negligible. So above equation can be written as $\lambda = \frac{h}{\sqrt {2m_{\circ} K}} \qquad(

Magnetic field at the center of circular loop

Mathematical Analysis of magnetic field at the center of circular loop: Let us consider, a current-carrying circular loop of radius $a$ in which $i$ current is flowing. Now take a small length of a current element $dl$ so magnetic field at the center of a circular loop due to the length of current element $dl$. According to Biot-Savart Law: $dB=\frac{\mu_{\circ}}{4 \pi} \frac{i .dl .sin\: \theta}{a^{2}} $ Here $\theta$ is the angle between length of current element $\left( \overrightarrow{dl} \right)$ and radius $\left( \overrightarrow{a} \right)$. These are perpendicular to each other i.e. $\theta = 90^{\circ}$ $dB=\frac{\mu_{\circ}}{4 \pi} \frac{i .dl .sin\: 90^{\circ}}{a^{2}} $ $dB=\frac{\mu_{\circ}}{4 \pi} \frac{i .dl }{a^{2}} \qquad \left(1 \right)$ The magnetic field at the center due to a complete circular loop $B=\int dB \qquad \left(2 \right)$ From equation $(1)$ and equation $(2)$ $B=\int \frac{\mu_{\circ}}{4 \pi} \frac{i .dl}{a^{2}}$ $B

Ampere's Circuital Law and its Modification

Ampere's Circuital Law Statement: When the current flows in any infinite long straight conductor then the line integration of the magnetic field around the current-carrying conductor is always equal to the $\mu_{0}$ times of the current. $\int \overrightarrow{B}. \overrightarrow{dl} = \mu_{\circ} i$ Derivation of Ampere's Circuital Law: Let us consider, An infinite long straight conductor in which $i$ current is flowing, then the magnetic field at distance $a$ around the straight current carrying conductor $B=\frac{\mu_{\circ}}{2 \pi} \frac{i}{a}$ Now the line integral of the magnetic field $B$ in a closed loop is $\oint \overrightarrow{B}. \overrightarrow{dl} = \oint \frac{\mu_{\circ}}{2 \pi} \frac{i}{a} dl$ $\oint \overrightarrow{B}. \overrightarrow{dl} = \frac{\mu_{\circ}}{2 \pi} \frac{i}{a} \oint dl$ $\oint \overrightarrow{B}. \overrightarrow{dl} = \frac{\mu_{\circ}}{2 \pi} \frac{i}{a} \left(2 \pi a \right) \qquad \left( \because

Preparation of nanostructured particles by sol-gel method

Introduction: The advent of the sol-gel process occurred in the year $1921$. In the $1960s$, its development was given due to the need for new synthesis methods in the nuclear industry. The sol-gel method is a widely used wet chemical technique to fabricate nanostructured materials. This technique is used to prepare nanoparticles of ceramics, glassy, and composite materials at relatively low temperatures based on wet chemical processing. It involves the conversion of a precursor solution (i.e. sol) into a solid three-dimension network (i.e. gel) through hydrolysis and condensation reactions of precursor compounds. There are following steps are given below to fabricate nanostructured material through the sol-gel method. 1. Precursor Compound Selection: The first step involves selecting the appropriate precursor compounds, usually metal alkoxides or inorganic salts, that will form the desired material upon hydrolysis and condensation. These precursors should be

Synthesis of Nanomaterials

In the field of nanomaterials, there are two main approaches to their synthesis and fabrication. A.) Top-Down Approach B.) Bottom-up Approach These two approaches are based on the methods used to create or assemble nanoscale materials and structures. A.) Top-Down Approach: In this approach, large-scale materials (i.e. Bulk materials) are broken down into smaller and smaller components until they reach the desired nanoscale dimensions i.e. firstly the bulk material is converted into powder form and then the powder's form is converted into nanoparticles. There are various physical methods (like Arc discharge method, Electron beam lithography, Mechanical grinding, etc) used to convert the bulk material into powder form and powder form is converted into nanoparticles by chemical methods (like Sol-Gel Process, Electrochemical method, Microemulsion etc.). One of the challenges with the top-down approach is that it may lead to a lack of control over the final nanomaterial'