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**Hint:**In order to answer the question we will first find out the radians per Second by the formula ${\omega _r} = \dfrac{1}{{\sqrt {LC} }}$ , then we will find out the Q-Factor of the series by using the formula $Q = \dfrac{{{\omega _r}L}}{R}$ and finally we will find out the improved sharpness of the resonance of the circuit according to the given condition.

**Complete answer:**

The Q factor, which describes how quickly energy decays in an oscillating system, is used to define the sharpness of resonance. For a rise or decrease in damping, the sharpness of resonance increases or decreases, and as the amplitude increases, the sharpness of resonance decreases.

Inductance, \[L{\text{ }} = {\text{ }}3.0{\text{ }}H\]

Capacitance, \[C{\text{ }} = {\text{ }}27{\text{ }}\mu F{\text{ }} = {\text{ }}27{\text{ }} \times {\text{ }}{10^{ - 6}}C\]

Resistance, \[R{\text{ }} = {\text{ }}7.4{\text{ }}\Omega \] At resonance, angular frequency of the source for the given LCR series circuit is given as:

${\omega _r} = \dfrac{1}{{\sqrt {LC} }}$

$

{\omega _r} = \dfrac{1}{{\sqrt {3 \times 27 \times {{10}^{ - 6}}} }} \\

{\omega _r} = \dfrac{{{{10}^3}}}{9} \\

$

${\omega _r} = 111.11\,rad\,{s^{ - 1}}$

Q-Factor of the series:

$Q = \dfrac{{{\omega _r}L}}{R}$

$Q = \dfrac{{111.11 \times 3}}{{7.4}} = 45.0446$

We need to reduce \[R\]to half, i.e., Resistance, to increase the sharpness of the resonance by reducing its ‘full width at half limit' by a factor of \[2\] without modifying

$ = \dfrac{R}{2} = \dfrac{{7.4}}{2} = 3.7\Omega $

**$\therefore $ For improvement in sharpness of resonance by a factor of $2$ ,Q should be doubled . To double Q with changing ${\omega _r}$, R should be reduced to half ,i.e., to $3.7\Omega $**

**Note:**The quality factor is a ratio of resonant frequency to bandwidth, and the higher the circuit \[Q,\]the smaller the bandwidth, \[Q{\text{ }} = {\text{ }}{{\text{f}}_r}{\text{ }}/BW\]. During each phase of oscillation, it compares the maximum or peak energy stored in the circuit (the reactance) to the energy dissipated (the resistance)..

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