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Showing posts from November, 2022

### Relation between angular velocity and linear velocity

Relation between angular velocity $(\omega)$ and linear velocity$(v)$: We know that the angular displacement of the particle is $\Delta \theta= \frac{\Delta s}{r} \qquad(1)$ Where $r$ = The radius of a circle. Now divide by $\Delta t$ on both side of equation $(1)$ $\frac{\Delta \theta}{\Delta t}=\frac{1}{r} \frac{\Delta s}{\Delta t}$ If $\Delta t \rightarrow 0$ then the above equation can be written as $\underset{\Delta t \rightarrow 0}{Lim}\: \frac{\Delta \theta}{\Delta t}=\frac{1}{r}\: \underset{\Delta t \rightarrow 0}{Lim} \: \frac{\Delta s}{\Delta t} \qquad(2)$ Where $\underset{\Delta t \rightarrow 0}{Lim}\: \frac{\Delta \theta}{\Delta t}$ = Instantaneous Angular Velocity $(\omega)$ $\underset{\Delta t \rightarrow 0}{Lim} \: \frac{\Delta s}{\Delta t}$= Instantaneous Linear Velocity $(v)$ Now equation $(2)$ can be written as $\omega=\frac{1}{r}v$ $v=r\omega$ This is the relation between linear velocity and angular velocity.

### Conservation's Law of linear momentum and its Derivation

Derivation of Conservation's law of Linear momentum from Newton's Second Law and Statement: Let us consider, A particle that has mass $m$ and is moving with velocity $v$ then According to Newton's second law the applied force on a particle is Derivation From General Form Derivation from Differential Form $F=ma$ $F=m \frac{\Delta v}{\Delta t}$ $F= \frac{\Delta (mv)}{\Delta t}$ $F= \frac{\Delta P}{\Delta t} \qquad \left( \because P=mv \right)$ If the applied force on a body is zero then $\frac{\Delta P}{\Delta t}=0$ $\Delta P=0$ Here $\Delta P$ is change in momentum of the particle .i.e $P_{2}-P_{1}=0$ $P_{2}=P_{1}$ $P=Constant$ $F=ma$ $F=m \frac{dv}{dt}$ $F= \frac{dmv}{dt}$ $F= \frac{dP}{dt} \qquad \left( \because P=mv \right)$ If the applied force on a body is zero then $\frac{dP}{dt}=0$ $dP=0$ On integrating the above equation

### Alternating Current Circuit containing Capacitance only (C-Circuit)

Alternating Current Circuit Containing Capacitance Only (C-Circuit): Let us consider, A circuit containing a capacitor of capacitance $C$ only which is connected with an alternating EMF i.e electromotive force source i.e. Let us consider, A circuit containing a coil of inductance $L$ only which is connected with an alternating EMF i.e electromotive force source i.e. $E=E_{\circ}sin\omega t\qquad(1)$ When alternating emf is applied across the capacitor plates then the charge on capacitor plates varies continuously and correspondingly current flows in the connecting leads. Let the charge on the capacitor plates is $q$ and the current in the circuit at any instant is $i$. Since there is no resistance in the circuit then the instantaneous potential difference is $\frac{q}{C}$ across the capacitor plates must be equal to the applied emf i.e. $\frac{q}{C} = E_{\circ} sin \omega t$ $q = CE_{\circ} sin \omega t \qquad(2)$ The instantaneous current $i$ in the circuit is,

### Work done by a rotating electric dipole in uniform electric field

Derivation of Work done by a rotating electric dipole in uniform electric field : Let us consider, An electric dipole AB, made up of two charges $+q$ and $-q$, is placed at a very small distance $2l$ in a uniform electric field $\overrightarrow{E}$. If dipole $AB$ rotates at angle $θ$ from its equilibrium position. If $A'$ and $B'$ are the new position of a dipole in the electric field. Then force on $+q$ charge particle due to electric field→ $\overrightarrow{F_{+q}}=q\overrightarrow{E}\qquad (1)$ Then force on $-q$ charge particle due to electric field→ $\overrightarrow{F_{-q}}=q\overrightarrow{E}\qquad(2)$ Work done by rotating an electric dipole   So work done by a force on $+q$ charge particle to bring from position $A$ to position $A'$→ $\overrightarrow{W_{+q}}=\overrightarrow{F_{+q}}·\overrightarrow{AC}$ $\overrightarrow{W_{+q}}=q\overrightarrow{E}· \overrightarrow{AC}\qquad(3)$ Similarly, work done by the forc

### Electric field Intensity due a uniformly charged Spherical Shell

Electric field intensity at a different point in the field due to the uniformly charged spherical shell: Let us consider, a spherical shell of radius $R$ in which $+q$ charge is distributed uniformly on the surface of the sphere. Now find the electric field intensity at different points due to the spherical shell. These different points are: Electric field intensity at an external point of the spherical shell Electric field intensity on the surface of the spherical shell Electric field intensity at an internal point of the spherical shell 1. Electric field intensity at an external point of the spherical shell: If point $O$ is the center of the spherical shell, The electric field at the outside of the spherical shell can be determined by the following steps → First, take the point $P$ outside the sphere Draw a spherical surface of radius r which p