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Showing posts from October, 2023

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