Question

In the LCR circuit inductance is changed from $L$ to \[\dfrac{L}{2}\], to keep the same resonance frequency $C$ should be changed to
A. $2 \times C$
B. $\dfrac{C}{2}$
C. $4 \times C$
D. \[\dfrac{C}{4}\]

Answer 1

Hint: An LCR circuit is an electronic circuit having resistor inductor and capacitor in either series or parallel. This circuit acts as an harmonic oscillator for current. At resonant condition, the inductive reactance \[{X_L}\] will become equal to the capacitive reactance ${X_C}$.

Complete step by step answer:
In an LCR circuit,at resonant frequency the peak current will be maximum ,the current will act as a purely resistive circuit and the resultant for net reactance will be zero. At resonant frequency the voltage and current also will be in the same phase.
The total impedance acting on a LCR circuit is given by the equation
\[Z = \sqrt {{R^2} + {{({X_C} - {X_L})}^2}} \]
${X_C}$ and ${X_L}$ indicate the capacitive reactance and inductive reactance of the circuit. The value of inductance reactance ${X_L} = \omega L$ and that of capacitive reactance ${X_C} = \dfrac{1}{{\omega C}}$

At resonant frequency condition, the peak current will be maximum i. e. impedance will be minimum
$\omega \times L = \dfrac{1}{{\omega \times C}}$
Substituting $\omega = 2 \times \pi \times f$and rearranging the equation we get $f = \dfrac{1}{{2\pi \sqrt {LC} }}$. Here we can see that as long as $L \times C$ remain constant the resonant frequency is the same.
As $L$ becomes $\dfrac{L}{2}$ ,$C$ becomes $2 \times C$

Note: Depending on the value of the ${X_C} - {X_L}$ the character of circuit changes.
If ${X_C}$>\[{X_L}\] the circuit is called capacitive.
If ${X_L}$>\[{X_C}\] the circuit is called inductive.
If ${X_L}$=${X_C}$ it is called a resonant circuit.
Some of the applications of resonant circuit include establishing a condition of stable frequency in circuits. Resonant circuits also serve as filters. Another example includes radio where we tune it into the required resonant frequency. Various types of filters are made like High pass filter, low pass filter, band-stop filter, band pass filter etc.