Question

Match the following:

Column I	Column II
(a) Resonant frequency	(i) $VI\cos \varphi$
(b) Quality factor	(ii) $\frac{1}{2}L{I}^2$
(c) Average power	(iii) $\frac{1}{\sqrt{LC}}$
(d) Impedance	(iv) $\sqrt{R^2+{(X_L-X_C)}^2}$
(e) Magnetic potential energy	(v) $\frac{-E}{\left(\frac{dI}{dt}\right)}$
(f) Coefficient of self-induction	(vi) $\frac{w_{0}L}{R}$

• A. $a\to iii,b\to v,c\to i,d\to iv,e\to vi,f\to ii$
• B. $a\to ii,b\to iv,c\to v,d\to iii,e\to vi,f\to i$
• C. $a\to ii,b\to iii,c\to i,d\to iv,e\to v,f\to vi$
• D. none of these

Answer 1

The correct option is A $a\to iii,b\to v,c\to i,d\to iv,e\to vi,f\to ii$

(i) Resonant Frequency:

For a series RLC AC circuit, resonance occurs with impedance of capacitor equals that of the inductor in magnitude.

$X_C=X_L$

$\frac{1}{{\omega }_{o}C}={\omega }_{o}L$

$⟹{\omega }_o=\frac{1}{\sqrt{LC}}$

(ii) Quality factor:

Quality factor is defined as the ratio of bandwidth to the center frequency. For a series RLC AC circuit, its value comes out to be equal to $Q=\frac{{\omega }_{o}L}{R}$

(iii)

For a series RLC AC circuit,

Average Apparent power is: $S=VI$

AverageReal Power is: $P=VI\cos \varphi$

AverageReactive power is: $Q=VI\sin \varphi$

(iv) Impedance:

For a series RLC circuit, impedance is given by:

$Z=R+j(X_L-X_C)$

Its magnitude is: $|Z|=\sqrt{R^2+(X_L-X_C{)}^2}$

(v)

Inductors store magnetic potential energy in them. For an inductor $L$ carrying a current $I$, potential difference is given by:

$V=L\frac{dI}{dt}$

Hence, instantaneous power is: $P=VI$

$P=LI\frac{dI}{dt}$

Energy stored in the inductor is: $E=\int Pdt$

$E=\int LIdI$

$=\frac{1}{2}L{I}^2$

(vi)

Inductance is the property of a conductor by which an emf is induced in it by a change of electric current through it. It is defined as the negative of ratio of the induced emf to the rate of change of current.

$L=\frac{E}{\left(\frac{dI}{dt}\right)}$