Question

\(L=50\,mH,\; C=100\,\mu F,\; R=50\,\Omega\). What does this circuit represent? (Assume \(\omega=200\,rad/s\))

• A. Inductive circuit
• B. Capacitive circuit
• C. Resonant circuit
• D. Purely resistive circuit

Hint:

For an LCR circuit, first calculate \[ X_L=\omega L \] and \[ X_C=\frac{1}{\omega C}. \] Then compare them: \[ \boxed{ \begin{aligned} X_L\gt X_C &\Rightarrow \text{Inductive},\\ X_L\lt X_C &\Rightarrow \text{Capacitive},\\ X_L=X_C &\Rightarrow \text{Resonance}. \end{aligned} } \] At resonance, \[ \boxed{\omega=\frac{1}{\sqrt{LC}}} \] and the impedance is equal to the resistance only.

Answer 1

The Correct Option is C

Concept: In a series LCR circuit, the nature of the circuit depends upon the comparison of the inductive reactance (\(X_L\)) and capacitive reactance (\(X_C\)). The reactances are given by \[ X_L=\omega L \] and \[ X_C=\frac{1}{\omega C}. \] The circuit behaves as: \[ \begin{aligned} X_L\gt X_C &\quad\Rightarrow\quad \text{Inductive circuit},\\[2mm] X_L\lt X_C &\quad\Rightarrow\quad \text{Capacitive circuit},\\[2mm] X_L=X_C &\quad\Rightarrow\quad \text{Resonant circuit}. \end{aligned} \] At resonance, \[ Z=R, \] the impedance becomes minimum, the current is maximum, and the phase difference between voltage and current is zero.

Step 1: Write the given data.
Given, \[ L=50\,mH=50\times10^{-3}\,H=0.05\,H, \] \[ C=100\,\mu F =100\times10^{-6}\,F =1\times10^{-4}\,F, \] \[ R=50\,\Omega, \] \[ \omega=200\,rad/s. \]

Step 2: Calculate the inductive reactance.
The inductive reactance is \[ X_L=\omega L. \] Substituting the given values, \[ X_L = 200\times0.05. \] Therefore, \[ \boxed{X_L=10\,\Omega.} \]

Step 3: Calculate the capacitive reactance.
The capacitive reactance is \[ X_C=\frac{1}{\omega C}. \] Substituting the values, \[ X_C = \frac{1}{200\times1\times10^{-4}}. \] Since, \[ 200\times10^{-4}=0.02, \] therefore, \[ X_C = \frac{1}{0.02} = 50\,\Omega. \] Thus, \[ \boxed{X_C=50\,\Omega.} \]

Step 4: Compare the reactances.
We have, \[ X_L=10\,\Omega, \] \[ X_C=50\,\Omega. \] Since, \[ X_C\gt X_L, \] the circuit behaves as a capacitive circuit. Hence, \[ \boxed{\textbf{Option (B)}}. \]

Important Observation: \[ X_L=X_C. \] Here, \[ 10\,\Omega\neq50\,\Omega, \] therefore the circuit cannot be resonant. The correct classification is \[ \boxed{\textbf{Capacitive Circuit}.} \]