Question

In an $LCR$ circuit, capacitance is changed from $C$ to $2C$. For the resonant frequency to remain unchanged, the inductance should be changed from $L$ to :
A. $4L$
B. $\dfrac{L}{2}$
C. $2L$
D. $\dfrac{L}{4}$

Answer 1

a LCR circuit is an electronic circuit which is a combination of capacitor, inductor and resistor which are either connected in series or in parallel. For the resonance condition inductive reactance becomes equal to capacitive reactance.

Formula used:
$X_L = X_C$
Where $X_L$= inductive reactance and $X_C$=capacitive reactance.

Complete step by step answer:
The resonance of a series LCR circuit occurs when the inductive and capacitive reactance are equal in magnitude but cancel each other because they are 180 degrees apart in phase. In resonance condition a circuit acts as a pure resistive circuit and has maximum current in it.
In LCR circuit, the impedance is given by:
${Z^2} = {R^2} + {\left( {X_L - X_C} \right)^2}$
where $R$ is resistance, $X_L$=​ inductive reactance and $X_C$=capacitive reactance.

At Resonance in LCR circuit,
$X_L = X_C$
$\Rightarrow \omega L = \dfrac{1}{{\omega C}}$
$\Rightarrow {\omega ^2} = \dfrac{1}{{LC}}$
$\Rightarrow \omega = \dfrac{1}{{\sqrt {LC} }}$
In the resonance condition the frequency remains unchanged. Hence $\sqrt {LC}$=constant. It means that even $LC$ is constant. Hence we can write that :
${L_1}{C_1} = {L_2}{C_2}$
Where $L_2$ is inductance initially and $L_2$ is final inductance.
We know that here ${C_1} = C$ and ${C_2} = 2C$ let us now substitute these values, we get:
${L_1}C = {L_2}2C$
$\Rightarrow {L_2} = \dfrac{{{L_1}C}}{{2C}}$
$\therefore {L_2} = \dfrac{{{L_1}}}{2}$

Hence the correct answer is option B.

Note: depending on the values of the reactance the behavior of the circuit changes, like: If $X_L > X_C$ the circuit is inductive in nature, if $X_L < X_C$, the circuit behaves like a capacitive circuit and if $X_L = X_C$. Also note that $X_L = \omega L$ and not $X_L = \dfrac{1}{{\omega L}}$ whereas $X_C = \dfrac{1}{{\omega C}}$ and not $X_C = \omega C$.