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Physical Significance of Maxwell's Equations

Physical Significance: The physical significance of Maxwell's equations obtained from integral form are given below: Maxwell's First Equation: 1. The total electric displacement through the surface enclosing a volume is equal to the total charge within the volume. 2. It represents Gauss Law. 3. This law is independent of time. Charge acts as source or sink for the lines of electric force. Maxwell's Second Equation: 1. The total magnetic flux emitting through any closed surface is zero. An isolated magnet do not exist monopoles. 2. There is no source or sink for lines of magnetic force. 3. This is time independent equation. Maxwell's Third Equation: 1. The electromotive force around the closed path is equal to the time derivative of the magnetic displacement through any surface bounded by the path. 2. This gives relation between electric field $E$ and magnetic induction $B$. 3. This expression is time varying i.e. $E$ is generate

Circuit containing Inductor and Capacitor in Series (L-C Series Circuit )

Mathematical Analysis of L-C Series Circuit : Let us consider, a circuit containing inductor $L$ capacitor $C$ and these are connected in series. If an alternating voltage source is applied across it then the resultant voltage of the L-C circuit $V=V_{L} - V_{C} \qquad(1)$ We know that: $V_{L} = iX_{L}$ $V_{C} = iX_{C}$ So from equation $(1)$ $V= iX_{L} - iX_{C} $ $V=i \left(X_{L} - X_{C} \right) $ $\frac{V}{i}=\left(X_{L} - X_{C} \right) $ $Z=\left(X_{L} - X_{C} \right) \qquad(2)$ Where $Z \rightarrow$ Impedance of L-C circuit. $X_{L} \rightarrow$ Inductive Reactance which has value $\omega L$ $X_{C} \rightarrow$ Capacitive Reactance which has value $\frac{1}{\omega C}$ So from equation $(2)$, we get $Z=\left( \omega L - \frac{1}{\omega C} \right) \qquad(3)$ The phase of resultant voltage: The phase of resultant voltage from current is $90^{\circ}$ as shown in the figure above. The Impedance and Phase at Re

Circuit containing Capacitor and Resistor in Series (C-R Series Circuit )

Mathematical Analysis of C-R Series Circuit : Let us consider, a circuit containing capacitor $C$ resistor $R$ and these are connected in series. If an alternating voltage source is applied across it then the resultant voltage of the C-R circuit $V=\sqrt{ V_{C} ^{2} + V^{2}_{R}} \qquad(1)$ We know that: $V_{R} = iR$ $V_{C} = iX_{C}$ So from equation $(1)$ $V=\sqrt{\left( iX_{C} \right)^{2} + \left(iR\right)^{2}} $ $V=i\sqrt{\left( X_{C} \right)^{2} + R^{2}} $ $\frac{V}{i}=\sqrt{\left( X_{C} \right)^{2} + R^{2}} $ $Z=\sqrt{\left( X_{C} \right)^{2} + R^{2}} \qquad(2)$ Where $Z \rightarrow$ Impedance of C-R circuit. $X_{C} \rightarrow$ Capacitive Reactance which has value $\frac{1}{\omega C}$ So from equation $(2)$, we get $Z=\sqrt{\left( \frac{1}{\omega C} \right)^{2} + R^{2}} \qquad(3)$ The phase of resultant voltage: If the phase of resultant voltage from from current is $\phi$ then $tan \phi = \frac{X_{C} }{R}

Circuit containing Inductor and Resistor in Series (L-R Series Circuit )

Mathematical Analysis of L-R Series Circuit : Let us consider, a circuit containing inductor $L$ resistor $R$ and these are connected in series. If an alternating voltage source is applied across it then the resultant voltage of the L-R circuit $V=\sqrt{ V_{L} ^{2} + V^{2}_{R}} \qquad(1)$ We know that: $V_{R} = iR$ $V_{L} = iX_{L}$ So from equation $(1)$ $V=\sqrt{\left( iX_{L} \right)^{2} + \left(iR\right)^{2}} $ $V=i\sqrt{\left( X_{L} \right)^{2} + R^{2}} $ $\frac{V}{i}=\sqrt{\left( X_{L} \right)^{2} + R^{2}} $ $Z=\sqrt{\left( X_{L} \right)^{2} + R^{2}} \qquad(2)$ Where $Z \rightarrow$ Impedance of L-R circuit. $X_{L} \rightarrow$ Inductive Reactance which has value $\omega L$ So from equation $(2)$, we get $Z=\sqrt{\left( \omega L \right)^{2} + R^{2}} \qquad(3)$ The phase of resultant voltage: If the phase of resultant voltage from from current is $\phi$ then $tan \phi = \frac{X_{L} }{R} \qquad(4)$ $tan

Circuit containing Inductor, Capacitor, and Resistor in Series (L-C-R Series Circuit )

Mathematical Analysis of L-C-R Series Circuit : Let us consider, a circuit containing inductor $L$, capacitor $C$, and resistor $R$ and these are connected in series. If an alternating voltage source is applied across it then the resultant voltage of the L-C-R circuit $V=\sqrt{\left( V_{L} -V_{C} \right)^{2} + V^{2}_{R}} \qquad(1)$ We know that: $V_{R} = iR$ $V_{L} = iX_{L}$ $V_{C} = iX_{C}$ So from equation $(1)$ $V=\sqrt{\left( iX_{L} - iX_{C} \right)^{2} + (iR)^{2}} $ $V=i\sqrt{\left( X_{L} - X_{C} \right)^{2} + R^{2}} $ $\frac{V}{i}=\sqrt{\left( X_{L} - X_{C} \right)^{2} + R^{2}} $ $Z=\sqrt{\left( X_{L} - X_{C} \right)^{2} + R^{2}} \qquad(2)$ Where $Z \rightarrow$ Impedance of L-C-R circuit. $X_{L} \rightarrow$ Inductive Reactance which has value $\omega L$ $X_{C} \rightarrow$ Inductive Reactance which has value $\frac{1}{\omega C}$ So from equation $(2)$, we get $Z=\sqrt{\left( \omega L - \frac{1}{\omega C} \right)^{2}

Merits and Demerits of Alternating Current in Comparison to Direct Current

Merits and Demerits of AC in Comparison to DC (a) Merits (i) Alternating current can be increased or decreased by using a transformer. This is the reason that Alternating current can be transmitted from one place to other place at relatively lower expenditure and minimum loss of energy. In Direct current, it is not possible. (ii) Alternating current can be controlled by choke coil or capacitor at very small loss of energy. To control Direct current resistance is required in which energy loss is very high. (ii) Alternating current can easily be converted into Direct current by using a rectifier but converting Direct current into Alternating current is not easy. (iv) Alternating current is cheaper than DC. (life of a cell or battery is very limited). (b) Demerits (i) Alternating current is more dangerous as compared to Direct current. (ii) Alternating current cannot be used in electrolysis. (iii) Most of the Alternating current of high frequency flows on the su

Power in Alternating Current Circuit

Definition of Power in Alternating Current Circuit: The rate of power consumption in an alternating current circuit is known as power in the circuit. Mathematical Analysis: Let us consider an alternating current circuit in which the voltage and current at any instant are given by $V=V_{\circ} sin \omega t \qquad(1)$ $i=i_{\circ} sin \left( \omega t - \phi \right) \qquad(2)$ Where $\phi$ $\rightarrow$ Phase difference between voltage and current So instantaneous Power $P=Vi$ $P=\left\{V_{\circ} sin \omega t \right\} \left\{ i_{\circ} sin \left( \omega t - \phi \right) \right\}$ $P=V_{\circ} i_{\circ} \: sin \omega t \: sin \left( \omega t - \phi \right)$ $P=\frac{V_{\circ} i_{\circ}}{2} \left[2\: sin \: \omega t \: sin \left( \omega t - \phi \right)\right]$ $P=\frac{V_{\circ} i_{\circ}}{2} \left[ cos \left( \omega t -\omega t + \phi \right) - cos \left( \omega t + \omega t - \phi \right) \right]$ $P=\frac{V_{\circ} i_{\circ}}{2} \left[ cos \left(

Self Induction Phenomenon and its Coefficient

Self Induction: When a changing current flows in a coil then due to the change in magnetic flux in the coil produces an electro-motive force $\left(emf \right)$ in that coil. This phenomenon is called the principle of Self Induction. The direction of electro-motive force can be found by applying "Lenz's Law". Mathematical Analysis of Coefficient of Self Induction: Let us consider that a coil having the number of turns is $N$. If the change in current is $i$, then linkage flux in a coil will be $N \phi \propto i$ $N\phi = L i \qquad(1)$ Where $L$ $\rightarrow$ Coefficient of Self Induction. According to Faraday's law of electromagnetic induction. The electro-motive force $\left(emf \right)$ in a coil is $e=-N\left( \frac{d \phi}{dt} \right)$ $e=-\frac{d \left(N \phi \right)}{dt} \qquad(2)$ From equation $(1)$ and equation $(2)$ $e=-\frac{d \left(L i\right)}{dt} $ $e=-L \left(\frac{d i}{dt} \right) $ $L = \frac{e}{\

Mutual Induction Phenomenon and its Coefficient

Mutual Induction: When two coils are placed near each other then the change in current in one coil ( Primary Coil) produces electro-motive force $\left( emf \right)$ in the adjacent coil ( i.e. secondary coil). This phenomenon is called the principle of Mutual Induction. The direction of electro-motive force $\left( emf \right)$ depends or can be found by "Lenz's Law" Mathematical Analysis of Coefficient of Mutual Induction: Let us consider that two coils having the number of turns are $N_{1}$ and $N_{2}$. If these coils are placed near to each other and the change in current of the primary coil is $i_{1}$, then linkage flux in the secondary coil will be $N_{2}\phi_{2} \propto i_{1}$ $N_{2}\phi_{2} = M i_{1} \qquad(1)$ Where $M$ $\rightarrow$ Coefficient of Mutual Induction. According to Faraday's law of electromagnetic induction. The electro-motive force $\left( emf \right)$ in the secondary coil is $e_{2}=-N_{2}\left( \frac{d \phi_

Faraday's laws of electromagnetic induction

Faraday's Laws of Electromagnetic Induction: The Faraday's experiment shows the two laws which are known as Farday's laws of electromagnetic induction First Law (Neumann's Law): The rate of change of magnetic flux through a circuit is equal to the emf produced in the circuit. This is also known as " Neumann Law " $e=-\frac{\Delta \phi}{ \Delta t}$ Here negative sign shows the direction of emf. If $\Delta t \rightarrow 0$ $e=-\frac{d \phi}{ d t}$ This equation represents an independent experimental law that cannot be derived from other experimental laws. If the circuit is a tightly wound coil of $N$ turns, then the induced emf $e=-N\frac{d \phi}{ d t}$ $e=-\frac{d \left(N \phi\right)}{ dt}$ Here $N \phi$ is called the 'Linkage magnetic flux'. Note: The change in flux induces emf, not the current. Second Law (Lenz's Law): The direction of induced EMF produced in a closed circuit is such that it opposes th

Paramagnetic Substances and Its properties

Paramagnetic Substances : Those substances, which are placed in the external magnetic field and they are weakly magnetized in the direction of the external magnetic field, are called paramagnetic substances. The susceptibility $\chi_{m} $ of paramagnetic substances is small and positive. Further, When a paramagnetic substance is placed in the magnetic field, then the flux density of the paramagnetic substance is slightly more than the free space. Thus, the relative permeability of paramagnetic substance $\mu_{r}$, is slightly more than 1. Properties of Paramagnetic substances: 1. When a rod of a paramagnetic material is suspended freely between external magnetic poles (i.e. Between North and South Poles) then its axis becomes along the direction of the external magnetic field $B$ (Figure). The poles produced on the two sides of the rod are opposite to the poles of the external magnetic field. 2. In a non-uniform magnetic field, a paramagnetic substance tends to move fro