Name: Magnetic Energy Density Stored in a Solenoid
Uploaded: 2019-07-19
Duration: 6 min 18 s
Description: The fourth state of matter is plasma and it is superheated matter. Where can we store this kind of energy? In a solenoid! Tune in to learn more about energy...

Question

Obtain an expression for impedance of resistor, pure inductor and capacitor connected in series across alternating e.m.f. State formula for phase difference.

Accepted Answer

Consider an alternating e.m.f. applied to a series combination of pure inductor of inductance $L$ , capacitor of capacitance $C$ and resistor of resistance $R$ . Let $e_{L}, e_{C}$ and $e_{R}$ be the r.m.s. voltages across inductor, capacitor and resistor respectively. Let $i$ be the r.m.s. current flowing through the circuit. As $e_{R}$ is in phase with current $i$ , the vector $e_{R}$ is drawn in the same direction as that of $i$ along positive direction of $X -$ axis.

$e_{L}$ leads the current by $\frac{π}{2}$ radian, hence vector $e_{L}$ is drawn in the positive direction of $Y$ axis, and similarly, $e_{C}$ lags the current by $\frac{π}{2}$ radian, so the vector $e_{C}$ is drawn in the negative direction of the $Y$ axis. The voltages $e_{L}, e_{C}$ and $e_{R}$ are given by,

$e_{R} = i R, e_{L} = i X_{L} & e_{C} = i X_{C}$

Let $e_{L} > e_{C}$ . the resultant voltage $e_{L} - e_{C}$ is in the direction of $e_{C}$ as shown in phasor diagram. Resultant of $e_{R}, e_{L} & e_{C}$ gives the r.m.s. value $(e)$ of applied e.m.f. From phasor diagram,
Hence, $e^{2} = {e_{R}}^{2} + {(e_{L} - e_{C})}^{2} = {(i R)}^{2} + {(i X_{L} - i X_{C})}^{2} = i^{2} [R^{2} + {(X_{L} - X_{C})}^{2}]$
$∴ e = i \sqrt{R^{2} + {(X_{L} - X_{C})}^{2}} = i Z$
Hence, the net impedance of the circuit is given by, $Z = \sqrt{R^{2} + {(X_{L} - X_{C})}^{2}}$
From the phase diagram, it can be observed that, $\tan ϕ = \frac{e_{L} - e_{C}}{e_{r}} = \frac{ω L - \frac{1}{ω C}}{R}$ or $ϕ = \tan^{- 1} (\frac{ω L - \frac{1}{ω C}}{R})$

Obtain an expression for impedance of resistor, pure inductor and capacitor connected in series across alternating e.m.f. State formula for phase difference.

Important Questions on Electromagnetic Induction and Alternating Current

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