Question

Match the following:

$\text{Column I}$	$\text{Column II}$
$(i)\; \text{Resonant}$ $\text{frequency}$	$(a)VI\cos \varphi$
$(ii)\; \text{Quality factor}$	$(b)\dfrac{1}{2}L{{I}^{2}}$
$(iii)\; \text{Average power}$	$(c)\dfrac{1}{\sqrt{LC}}$
$(iv)\; \operatorname{Impedance}$	$(d)\sqrt{{{R}^{2}}+{{({{X}_{L}}-{{X}_{C}})}^{2}}}$
$(v)\; \text{Magnetic potential energy}$	$(e)\dfrac{E}{(\dfrac{dI}{dt})}$
$(vi)\; \text{Coefficient of self induction}$	$(f)\dfrac{{{\omega }_{0}}L}{R}$

$\begin{aligned} & A)a\to iii,b\to v,c\to i,d\to iv,e\to vi,f\to ii \\\ & B)a\to ii,b\to iv,c\to v,d\to iii,e\to vi,f\to i \\\ & C)a\to ii,b\to iii,c\to i,d\to iv,e\to v,f\to vi \\\ & D)\text{None of these} \\\ \end{aligned}$

Answer 1

Hint: Students can use the dimensional analysis to compare the results in both the columns. The physical meaning of the terms in $\text{Column I}$can also be taken into consideration in order to find their formulae in $\text{Column II}$.

Complete step by step answer:
In order to match the above columns let us first take a look into the meaning or definition of the terms given in $\text{Column I}$ and then compare them with those given in $\text{Column II}$.

Definitions:
(i) Resonant frequency: Resonance is the phenomenon of increase of the amplitude of oscillation of a particular system. This increase in amplitude to the highest degree occurs at a particular frequency, which is known as the resonant frequency.
If ${{f}_{0}}$ is the resonant frequency of a system, then
${{f}_{0}}=\dfrac{1}{\sqrt{LC}}$

(ii) Quality factor: Quality factor in an LCR circuit is the ratio of the resonant frequency to the difference in two frequencies taken on both sides of the resonant frequency such that at each frequency, the current amplitude becomes $\dfrac{1}{\sqrt{2}}$ times the value at resonant frequency.
If $Q$ represents the quality factor of an LCR circuit, then
$Q=\dfrac{{{\omega }_{0}}L}{R}$
where, ${{\omega }_{0}}$ is the resonant frequency
$L$ is the self- inductance and
$R$ is the resistance of the circuit.

(iii) Average power: In an LCR circuit, if ${{P}_{av}}$ represents the average power of the circuit, then,
$\begin{aligned} & {{P}_{av}}=\text{virtual emf}\times \text{virtual current}\times \cos \phi \\\ & \Rightarrow {{P}_{av}}=V\times I\times \cos \phi \\\ \end{aligned}$
where, $\cos \phi$ is known as the power factor.

(iv) Impedance: The effective resistance in the LCR circuit which opposes the flow of current through it is known as impedance.
$Z=\sqrt{{{R}^{2}}+{{({{X}_{L}}-{{X}_{C}})}^{2}}}$
Where, $Z$ is the impedance, $R$ is the resistance, ${{X}_{L}}$ is the inductive reactance and ${{X}_{C}}$ is the capacitive reactance of the circuit.

(v) Magnetic potential energy: The energy stored in a magnetic field induced in an inductor is equal to the work required to produce a current across the inductor.
If $M$ represents the magnetic potential energy across an inductor circuit, and $I$ represents the current through the circuit then,
$M=\dfrac{1}{2}L{{I}^{2}}$

(vi) Coefficient of self induction: Self-inductance of a coil is numerically equal to the change in the magnetic flux linked with the coil when unit current passes through it.
If $L$ is the coefficient of self induction or self inductance, then,
$L=\dfrac{E}{(\dfrac{dI}{dt})}$
where, $E$is the emf of the circuit and $\dfrac{dI}{dt}$ is the change in current across the circuit.
Therefore, the answer is option $A)a\to iii,b\to v,c\to i,d\to iv,e\to vi,f\to ii$.

Note: The option for resonant frequency given in the question is in $rad/s$ , that is, it is the angular resonant frequency. In case of frequency, a term $2\pi$ should have appeared in the denominator.