In a series $LCR$ circuit,the capacitance is changed from $C$ to $4C$. To keep the resonance frequency unchanged,the new inductance should be:

$A$ resistor of $2 \ \Omega$,an inductor of $100 \ \mu H$,and a capacitor of $400 \ pF$ are connected in series across an $A$.$C$. source of $e_{rms} = 0.1 \ V$. At resonance,the voltage drop across the inductor is: (in $V$)

In a series $LCR$ resonant circuit,$R = 800 \ \Omega$,$C = 2 \ \mu F$,and the voltage across the resistance is $200 \ V$. The angular frequency is $250 \ rad/s$. At resonance,the voltage across the inductance is: (in $V$)

In the circuit shown in the figure,an a.c. source gives voltage $V = 20 \cos (2000 t)$. The impedance and r.m.s. current respectively will be:

$A$ telegraph line of length $100 \, km$ has a capacity of $0.01 \, \mu F/km$ and it carries an alternating current at $0.5 \, kHz$. If minimum impedance is required, then the value of the inductance that needs to be introduced in series is . . . . . . $mH$. (Take $\pi = \sqrt{10}$)

$A$ $50 \text{ Hz}$ $AC$ circuit has a $10 \text{ mH}$ inductor and a $2 \text{ } \Omega$ resistor in series. The value of capacitance to be placed in series in the circuit to make the circuit power factor as unity is