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

Hint: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 \[X_C = \omega C\].